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Definite Integrals: Integration by Substitution

Related Topics

Wize University Calculus 1 Textbook > Integrals

Integration by Substitution (U-Substitution)

8 Activities

Wize University Calculus 1 Textbook > Integrals

Properties of Definite Integrals

3 Activities

If ∫−27g(x)dx=13\int_{-2}^7g\left(x\right)dx=13 and ∫572g(x)=10\int_5^72g\left(x\right)=10, then ∫−152g(2x)dx=\int_{-1}^{\frac{5}{2}}g\left(2x\right)dx=
More Integration by Substitution (U-Substitution) Questions:
Integration by Substitution: Indefinite with Trig
Find the following integral: ∫1x2sin⁡(1x)cos⁡(1x) dx\displaystyle\int\frac{1}{x^2}\sin\left(\frac{1}{x}\right)\cos\left(\frac{1}{x}\right) \ dx.
Integration by Substitution: Indefinite with Power
Compute the following integral: ∫(x−6)(x+6)1/3 dx\displaystyle \int (x-6)(x+6)^{1/3}\,\text{d}x
Integration by Substitution
If ∫25f(2x) dx=6\int_2^5f\left(2x\right)\ dx=6 and ∫210f(x)=20\int_2^{10}f\left(x\right)=20, find ∫24f(x) dx\int_2^4f\left(x\right)\ dx.
Integration by Substitution
∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx
Integration by Substitution
Evaluate ∫eex+xdx\int_{ }^{ }e^{e^x+x}dx.
Integration by Substitution
Evaluate the definite integral ∫0π/3ln⁡(cos⁡x)tan⁡x dx{\displaystyle \int_{0}^{\pi/3}}\ln{(\cos x)}\tan x \space dx.
Practice Question: U-Sub *
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Practice Question
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Practice Question
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Practice Question
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Practice Question
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Practice Question: U-Sub *
Practice Question
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Practice Question: U-Sub *
Practice Question
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Integration by Substitution
Evaluate the definite integral ∫0π/3ln⁡(cos⁡x)tan⁡x dx{\displaystyle \int_{0}^{\pi/3}}\ln{(\cos x)}\tan x \space dx.
Integration by Substitution: Definite with Exponential
Compute the following integral: ∫13xe2x25 dx\displaystyle \int_{1}^{3}\frac{xe^{2x^2}}{5}\,\text{d}x
Integration by Substitution
Evaluate the definite integral ∫0π/3ln⁡(cos⁡x)tan⁡x dx{\displaystyle \int_{0}^{\pi/3}}\ln{(\cos x)}\tan x \space dx.
Substitution with Definite Integral
Practice: Substitution with Definite Integral
Evaluate ∫1e(ln⁡x−5)2xdx\displaystyle \int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Substitution with Definite Integral
Practice: Substitution with Definite Integral
Evaluate ∫1e(ln⁡x−5)2xdx\displaystyle \int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Integration by Substitution
∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx
Integration by Substitution
Evaluate the definite integral ∫0π/3ln⁡(cos⁡x)tan⁡x dx{\displaystyle \int_{0}^{\pi/3}}\ln{(\cos x)}\tan x \space dx.
Practice Question: U-Sub
Compute ∫ln⁡(x)x  ⁣dx.{\displaystyle\int}\frac{\ln(x)}{x}\de{x}.
Practice Question: U-Sub
Practice Question
Evaluate ∫sec⁡3xtan⁡3x(1+sec⁡3x)3/2  ⁣dx{\displaystyle\int}\frac{\sec 3x\tan 3x}{(1+\sec 3x)^{3/2}}\de{x}.
Integration by Substitution
Find the antiderivative of
f(x)=−cos⁡6x5sin⁡26xf(x)= -\frac{\cos 6x}{5\sin^2 6x}
Integration by Substitution
Evaluate ∫eex+xdx\int_{ }^{ }e^{e^x+x}dx.
Integration by Substitution
∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx
Integration by Substitution: Definite with Exponential
Compute the following integral: ∫13xe2x25 dx\displaystyle \int_{1}^{3}\frac{xe^{2x^2}}{5}\,\text{d}x
Substitution with Definite Integral
Evaluate ∫1e(ln⁡x−5)2xdx\displaystyle \int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Integration by Substitution
Evaluate the definite integral ∫0π/3ln⁡(cos⁡x)tan⁡x dx{\displaystyle \int_{0}^{\pi/3}}\ln{(\cos x)}\tan x \space dx.
Evaluate the integral ∫x3(1+x4)14dx\displaystyle \int_{ }^{ }x^3(1+x^4)^{\frac{1}{4}}dx.
Find the indefinite integral ∫x1+x2dx\int\frac{x}{1+x^2}dx.
Evaluate the integral I=∫cos⁡(x+1)x+1dx\displaystyle I=\int\frac{\cos(\sqrt{x+1})}{\sqrt{x+1}}dx.
Evaluate the integral ∫01x3⋅3x4+2 dx\displaystyle\int_0^1x^3\cdotp3^{x^4+2}\ dx.
Evaluate the integral ∫cos⁡x sec⁡2(sin⁡x)dx\displaystyle\int_{ }^{ }\cos x\ \sec^2\left(\sin x\right)dx.
Evaluate the integral ∫cos⁡x sec⁡2(sin⁡x)dx\displaystyle\int_{ }^{ }\cos x\ \sec^2\left(\sin x\right)dx.
Evaluate the integral ∫cos⁡x sec⁡2(sin⁡x)dx\displaystyle\int_{ }^{ }\cos x\ \sec^2\left(\sin x\right)dx.
Evaluate the integral ∫cos⁡x sec⁡2(sin⁡x)dx\displaystyle\int_{ }^{ }\cos x\ \sec^2\left(\sin x\right)dx.
Evaluate the integral ∫1arcsin⁡x1−x2dx\displaystyle\int_{ }^{ }\frac{1}{\arcsin x\sqrt{1-x^2}}dx. (If you need to input an inverse trigonometric function, the computer asks that you use the "arc" notation instead of "-1", i.e. arcsin⁡\arcsin instead of sin⁡−1\sin^{-1}.)
Find the definite integral ∫14exxdx\displaystyle\int_1^4\frac{e^{\sqrt{x}}}{\sqrt{x}}dx.
Practice Question: U-Sub
Compute ∫ln⁡(x)x  ⁣dx.{\displaystyle\int}\frac{\ln(x)}{x}\de{x}.
Complete the Square, Tan, Split the Fraction: Integration by Substitution
Evaluate ∫x(x2−x+1)2 dx\displaystyle\int\frac{x}{(x^2-x+1)^2} \ dx
Practice: Integration by Substitution (with back-substitution)
Practice: Integration by Substitution (with back-substitution)
Find ∫2x32x2+1dx\int_{ }^{ }\frac{2x^3}{2x^2+1}dx.
Integration by Substitution
∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx
Practice: Integration by Substitution
Practice: Integration by Substitution
Evaluate ∫02(x−2)e(x2−4x)dx\int_0^2\left(x-2\right)e^{\left(x^2-4x\right)}dx.
Practice: Integration by Substitution (~F2018 Final Q36)
Practice: Integration by Substitution
Find ∫cot⁡x ln⁡(sin⁡x)dx\int_{ }^{ }\cot x\ \ln\left(\sin x\right)dx.
Practice: Integration by Substitution (~F2017 Final Q35)
Practice: Integration by Substitution
Evaluate ∫1e(ln⁡x−5)2xdx\int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Practice: Property of Definite Integral
Practice: Property of Definite Integral
Suppose f(x)f(x) is continuous and let ∫07f(x)dx=7\displaystyle \int_0^7f\left(x\right)dx=7 and ∫710f(x)dx=10\displaystyle \int_7^{10}f\left(x\right)dx=10.
Find 5∫02f(5x)dx\displaystyle 5\int_0^2f\left(5x\right)dx.
Complete the Square, Tan, Split the Fraction: Integration by Substitution
Evaluate ∫x(x2−x+1)2 dx\displaystyle\int\frac{x}{(x^2-x+1)^2} \ dx
Integration by Substitution: Definite with Exponential
Evaluate the integral: ∫01x3 3x4+2 dx\displaystyle\int_0^1x^3 \ 3^{x^4+2}\ dx
Integration by Substitution: Powers and Root
Compute the following integral: ∫ x3x2+1 dx\displaystyle \int\,x^3\sqrt{x^2+1} \ \text{d}x
Practice: Indefinite with Root
Q:\textbf{Q:} Evaluate ∫2xx2−1dx\displaystyle\int_{ }^{ }\frac{2x}{\sqrt{x^2-1}}dx
Integration by Substitution: Indefinite with Root
Compute the following integral: ∫31−4x2 dx\displaystyle \int\frac{3}{\sqrt{1-4x^2}}\,\text{d}x
Integration by Substitution: Indefinite with Fraction
Compute the following integral: ∫11+2x2 dx\displaystyle \int \frac{1}{1+2x^2} \ \text{d}x
Practice: Indefinite with Inverse Trig
Q.\textbf{Q.} Compute the following integral: ∫arcsinx1−x2 dx\displaystyle \int\frac{\text{arcsin}{x}}{\sqrt{1-x^2}}\,\text{d}x
Integration by Substitution: Indefinite with Square Root
Compute the following integral: ∫3x+2 dx\displaystyle \int \sqrt{3x+2}\,\text{d}x
Evaluate ∫1e(ln⁡x−5)4x dx\displaystyle\int_1^e\frac{\left(\ln x-5\right)^4}{x}\ dx
Suppose f(x)f(x) is continuous and let ∫07f(x)dx=7\displaystyle\int_0^7f\left(x\right)dx=7 and ∫710f(x)dx=10 \displaystyle\int_7^{10}f\left(x\right)dx=10 , find 5∫02f(5x)dx\displaystyle5\int_0^2f\left(5x\right)dx.
Find ∫cot⁡x ln⁡(sin⁡x)dx\displaystyle\int_{ }^{ }\cot x\ \ln\left(\sin x\right)dx
Integration by Substitution: Indefinite with Trig
Find the following integral: ∫1x2sin⁡(1x)cos⁡(1x) dx\displaystyle\int\frac{1}{x^2}\sin\left(\frac{1}{x}\right)\cos\left(\frac{1}{x}\right) \ dx.
Integration by Substitution: Indefinite with Exponentials
Compute the following integral: ∫ez+ez dz\displaystyle \int e^{z+e^z}\,\text{d}z
Integration by Substitution: Indefinite with Power
Compute the following integral: ∫(x−6)(x+6)1/3 dx\displaystyle \int (x-6)(x+6)^{1/3}\,\text{d}x
Practice: Definite with Roots
Q.\textbf{Q.} Evaluate ∫14xx2+1 dx\displaystyle \int_{1}^{4}x\sqrt{x^2+1}\,\text{d}x
Integration by Substitution: Definite with Exponential
Compute the following integral: ∫13xe2x25 dx\displaystyle \int_{1}^{3}\frac{xe^{2x^2}}{5}\,\text{d}x
Practice: Indefinite with Inverse Trig
Compute the following integral: ∫arcsinx1−x2 dx\displaystyle \int\frac{\text{arcsin}{x}}{\sqrt{1-x^2}}\,\text{d}x
(a) =[arctan⁡(x)]22+C=\frac{[\arctan(x)]^2}{2}+C (b) =[arcsin⁡(x)]22+C=\frac{[\arcsin(x)]^2}{2}+C (c) =121−x2+C=\frac12\sqrt{1-x^2}+C (d) diverges
Practice: Integration by Substitution
Practice: Integration by Substitution derived from Final
Practice: Substitution with Definite Integral
Practice: Substitution with Definite Integral
Evaluate ∫1e(ln⁡x−5)2xdx\displaystyle \int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Practice: Integration by Substitution (with back-substitution)
Practice: Integration by Substitution (with back-substitution)
Find ∫2x32x2+1dx\int_{ }^{ }\frac{2x^3}{2x^2+1}dx.
Practice: Substitution with Definite Integral
Practice: Substitution with Definite Integral
Evaluate ∫1e(ln⁡x−5)2xdx\displaystyle \int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Example: Integration by Substitution
Evaluate ∫1x+x1/2dx\displaystyle \int_{ }^{ }\frac{1}{x+x^{1/2}}dx
Practice: Integration by Substitution (~2019 Test1 A10)
Practice: Integration by Substitution derived from Midterm
Evaluate ∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx
Practice: Substitution with Definite Integral
Practice: Substitution with Definite Integral
Evaluate ∫1e(ln⁡x−5)2xdx\displaystyle \int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Integration by Substitution
∫tan⁡x⋅ln⁡(cos⁡x)dx=\displaystyle\int_{ }^{ }\tan x\cdot\sqrt{\ln\left(\cos x\right)}dx=
Integration by Substitution
∫e1x3x2dx=\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{3x^2}dx=
Find ∫12exxdx\int_{ }^{ }\frac{1}{2e^{\sqrt{x}}\sqrt{x}}dx
Evaluate ∫−102x23−3x3dx\int_{-1}^0\frac{2x^2}{3-3x^3}dx.
Find ∫3x(x2−6)dx\displaystyle\int_{ }^{ }\frac{3x}{\sqrt{\left(x^2-6\right)}}dx.
Integration by Substitution
Evaluate the definite integral ∫0π/3ln⁡(cos⁡x)tan⁡x dx{\displaystyle \int_{0}^{\pi/3}}\ln{(\cos x)}\tan x \space dx.
Integration by Substitution
Evaluate the definite integral ∫0π/3ln⁡(cos⁡x)tan⁡x dx{\displaystyle \int_{0}^{\pi/3}}\ln{(\cos x)}\tan x \space dx.
Practice: Integration by Substitution (with back-substitution)
Practice: Integration by Substitution (with back-substitution)
Find ∫2x32x2+1dx\int_{ }^{ }\frac{2x^3}{2x^2+1}dx.
Substitution with Definite Integral
Practice: Substitution with Definite Integral
Evaluate ∫1e(ln⁡x−5)2xdx\displaystyle \int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Practice: Integration by Substitution (~2019 Test1 A10)
Practice: Integration by Substitution
Evaluate ∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx
Evaluate ∫sin⁡(ln⁡x)xdx\int_{ }^{ }\frac{\sin\left(\ln x\right)}{x}dx
Integration by substitution
Evaluate ∫02xx2−1dx\displaystyle\int_0^2 \dfrac{x}{x^2-1}dx
Practice: Property of Definite Integral
Suppose f(x) is continuous and let ∫07f(x)dx=7\int_0^7f\left(x\right)dx=7 and ∫710f(x)dx=10\int_7^{10}f\left(x\right)dx=10. Find 5∫02f(5x)dx5\int_0^2f\left(5x\right)dx
Integration by Substitution
Evaluate ∫eex+xdx\int_{ }^{ }e^{e^x+x}dx.
Integration by Substitution
Evaluate ∫13x2+1x3+3x dx\displaystyle \int_{1}^{3} \frac{x^2+1}{\sqrt{x^3+3x}} \,dx
Integration by substitution
Evaluate ∫01x31+x2 dx\displaystyle \int_{0}^{1} \frac{x^3}{\sqrt{1+x^2}} \,dx
Integration by Substitution
Find the antiderivative of
f(x)=−cos⁡6x5sin⁡26xf(x)= -\frac{\cos 6x}{5\sin^2 6x}
Practice Question: U-Sub
Practice Question
Compute ∫ln⁡(x)x  ⁣dx.{\displaystyle\int}\frac{\ln(x)}{x}\de{x}.
Evaluate ∫1xln⁡xdx\int_{ }^{ }\frac{1}{x\ln\sqrt{x}}dx
Find ∫x41+x10dx\int_{ }^{ }\frac{x^4}{1+x^{10}}dx
Evaluate ∫sin⁡(ln⁡x)xdx\int_{ }^{ }\frac{\sin\left(\ln x\right)}{x}dx
Evaluate ∫arctan⁡(1x)dx\int_{ }^{ }\arctan\left(\frac{1}{x}\right)dx
Evaluate ∫1xex+exdx\int_{ }^{ }\frac{1}{\sqrt{x}}e^{\sqrt{x}+e^{\sqrt{x}}}dx
Evaluate ∫12x2+23x3+2x+10dx\int_1^2\frac{x^2+\frac{2}{3}}{x^3+2x+10}dx
Evaluate ∫1ex−2dx\int_{ }^{ }\frac{1}{e^x-2}dx
Evaluate ∫1xln⁡xdx\int_{ }^{ }\frac{1}{x\ln\sqrt{x}}dx
∫141x+x3dx\int_1^4\frac{1}{\sqrt{x}+\sqrt{x^3}}dx
Practice Question: U-Sub *
Practice Question
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Practice Question: U-Sub
Practice Question
Evaluate ∫sec⁡3xtan⁡3x(1+sec⁡3x)3/2  ⁣dx{\displaystyle\int}\frac{\sec 3x\tan 3x}{(1+\sec 3x)^{3/2}}\de{x}.
Integration Medley
Practice Question
Evaluate ∫tan⁡xln⁡(cos⁡x)  ⁣dx{\displaystyle\int}\frac{\tan x}{\ln(\cos x)}\de{x}.
Practice Question: U-Sub
Evaluate
∫sec⁡3xtan⁡3x(1+sec⁡3x)3/2dx{\displaystyle\int}\frac{\sec 3x\tan 3x}{(1+\sec 3x)^{3/2}}dx
Practice: Property of Definite Integral
Suppose f(x) is continuous and let ∫07f(x)dx=7\int_0^7f\left(x\right)dx=7 and ∫710f(x)dx=10\int_7^{10}f\left(x\right)dx=10. Find 5∫02f(5x)dx5\int_0^2f\left(5x\right)dx
Evaluate ∫1xln⁡xdx\int_{ }^{ }\frac{1}{x\ln\sqrt{x}}dx
Evaluate ∫ex∗3exdx\displaystyle \int e^x*3^{e^x}dx
Practice Question: U-Sub
Evaluate
∫sec⁡3xtan⁡3x(1+sec⁡3x)3/2dx{\displaystyle\int}\frac{\sec 3x\tan 3x}{(1+\sec 3x)^{3/2}}dx
Evaluate ∫(cosx)2sin(x)dx\displaystyle \int (cosx)2^{sin(x)}dx
Evaluate∫ln2ln3ex(ex−1)3 dx\text{Evaluate} \displaystyle \int_{ln2}^{ln3} e^x(e^x-1)^3 \space dx
Find ∫1(x)(lnx)dx\displaystyle \int \frac{1}{(x)(lnx)}dx
Practice Question: U-Sub *
Compute∫1e3x−1  ⁣dx{\displaystyle\int}\frac{1}{\sqrt {e^{3x}-1}}\de{x}.
Practice Question: U-Sub
Evaluate ∫sec⁡3xtan⁡3x(1+sec⁡3x)3/2  ⁣dx{\displaystyle\int}\frac{\sec 3x\tan 3x}{(1+\sec 3x)^{3/2}}\de{x}.
Integration Medley
Evaluate ∫tan⁡xln⁡(cos⁡x)  ⁣dx{\displaystyle\int}\frac{\tan x}{\ln(\cos x)}\de{x}.
Practice Question: U-Sub
Compute ∫ln⁡(x)x  ⁣dx.{\displaystyle\int}\frac{\ln(x)}{x}\de{x}.
Find ∫1−3x3+6x−9x2dx\displaystyle\int\frac{1-3x}{\sqrt{3+6x-9x^2}}dx
Complete the Square, Tan, Split the Fraction: Integration by Substitution
Evaluate ∫x(x2−x+1)2 dx\displaystyle\int\frac{x}{(x^2-x+1)^2} \ dx
Evaluate the integral ∫cos⁡(x+1)x+1dx\displaystyle \int\frac{\cos(\sqrt{x+1})}{\sqrt{x+1}}dx. Use C for the constant of integration.
Example: Integration by Substitution
Evaluate ∫1x+x1/2dx\displaystyle \int_{ }^{ }\frac{1}{x+x^{1/2}}dx
Practice: Properties of Definite Integral
Evaluate ∫−π2π2xcos⁡2x sin⁡(ex2) dx\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}x\cos^2x\ \sin \left(e^{x^2}\right)\ dx.
Practice: Property of Definite Integral (~F2014 Final Q25) (~F2017 Final Q24)
Suppose f(x) is continuous and let ∫07f(x)dx=7\int_0^7f\left(x\right)dx=7 and ∫710f(x)dx=10\int_7^{10}f\left(x\right)dx=10. Find 5∫02f(5x)dx5\int_0^2f\left(5x\right)dx
Practice: Integration by Substitution
Evaluate ∫02(x−2)e(x2−4x)dx\int_0^2\left(x-2\right)e^{\left(x^2-4x\right)}dx.
Integration by Substitution
Find ∫cot⁡x ln⁡(sin⁡x)dx\int_{ }^{ }\cot x\ \ln\left(\sin x\right)dx.
Evaluate the integral I=∫cos⁡(x+1)x+1dx\displaystyle I=\int\frac{\cos(\sqrt{x+1})}{\sqrt{x+1}}dx.
Practice Question: U-Sub
Evaluate
∫sec⁡3xtan⁡3x(1+sec⁡3x)3/2dx{\displaystyle\int}\frac{\sec 3x\tan 3x}{(1+\sec 3x)^{3/2}}dx
Evaluate ∫01x31+x2 dx\displaystyle \int_{0}^{1} \frac{x^3}{\sqrt{1+x^2}} \,dx
Evaluate the definite integral ∫0π/3ln⁡(cos⁡x)tan⁡x dx{\displaystyle \int_{0}^{\pi/3}}\ln{(\cos x)}\tan x \space dx.
Evaluate∫ln2ln3ex(ex−1)3 dx\text{Evaluate} \displaystyle \int_{ln2}^{ln3} e^x(e^x-1)^3 \space dx
Evaluate ∫ex∗3exdx\displaystyle \int e^x*3^{e^x}dx
Practice: Indefinite with Inverse Trig
Q.\textbf{Q.} Compute the following integral: ∫arcsinx1−x2 dx\displaystyle \int\frac{\text{arcsin}{x}}{\sqrt{1-x^2}}\,\text{d}x
final114
Evaluate the integral: ∫e2x1+e2xdx\displaystyle\int_{ }^{ }\frac{e^{2x}}{1+e^{2x}}dx
Evaluate the integral ∫cos⁡(x+1)x+1dx\displaystyle \int\frac{\cos(\sqrt{x+1})}{\sqrt{x+1}}dx. Use C for the constant of integration.
Evaluate ∫1e(ln⁡x−5)4x dx\displaystyle\int_1^e\frac{\left(\ln x-5\right)^4}{x}\ dx
Find ∫cot⁡x ln⁡(sin⁡x)dx\displaystyle\int_{ }^{ }\cot x\ \ln\left(\sin x\right)dx
Suppose f(x)f(x) is continuous and let ∫07f(x)dx=7\displaystyle\int_0^7f\left(x\right)dx=7 and ∫710f(x)dx=10 \displaystyle\int_7^{10}f\left(x\right)dx=10 , find 5∫02f(5x)dx\displaystyle5\int_0^2f\left(5x\right)dx.
Integration by Substitution: Definite with Exponential
Evaluate the integral: ∫01x3 3x4+2 dx\displaystyle\int_0^1x^3 \ 3^{x^4+2}\ dx
Integration by Substitution: Indefinite with Trig
Find the following integral: ∫1x2sin⁡(1x)cos⁡(1x) dx\displaystyle\int\frac{1}{x^2}\sin\left(\frac{1}{x}\right)\cos\left(\frac{1}{x}\right) \ dx.
Integration by Substitution: Indefinite with Square Root
Compute the following integral: ∫3x+2 dx\displaystyle \int \sqrt{3x+2}\,\text{d}x
Integration by Substitution: Indefinite with Fraction
Compute the following integral: ∫11+2x2 dx\displaystyle \int \frac{1}{1+2x^2} \ \text{d}x
Practice: Definite with Roots
Q.\textbf{Q.} Evaluate ∫14xx2+1 dx\displaystyle \int_{1}^{4}x\sqrt{x^2+1}\,\text{d}x
Practice: Indefinite with Root
Q:\textbf{Q:} Evaluate ∫2xx2−1dx\displaystyle\int_{ }^{ }\frac{2x}{\sqrt{x^2-1}}dx
Integration by Substitution: Powers and Root
Compute the following integral: ∫ x3x2+1 dx\displaystyle \int\,x^3\sqrt{x^2+1} \ \text{d}x
Integration by Substitution: Indefinite with Exponentials
Compute the following integral: ∫ez+ez dz\displaystyle \int e^{z+e^z}\,\text{d}z
Integration by Substitution: Indefinite with Root
Compute the following integral: ∫31−4x2 dx\displaystyle \int\frac{3}{\sqrt{1-4x^2}}\,\text{d}x
Integration by Substitution: Indefinite with Power
Compute the following integral: ∫(x−6)(x+6)1/3 dx\displaystyle \int (x-6)(x+6)^{1/3}\,\text{d}x
Integration by Substitution: Definite with Exponential
Compute the following integral: ∫13xe2x25 dx\displaystyle \int_{1}^{3}\frac{xe^{2x^2}}{5}\,\text{d}x
Evaluate the integral ∫01x3⋅3x4+2 dx\displaystyle\int_0^1x^3\cdotp3^{x^4+2}\ dx.
Integration by Substitution
Evaluate
∫−ππ/2xcos⁡(2x2+1) dx\int_{-\pi}^{\pi/2} x \cos(2x^2+1)\ dx
Integration by Substitution
Evaluate
∫1e(ln⁡x−5)2xdx\displaystyle\int_1^e\dfrac{\left(\ln x-5\right)^2}{x}dx
Practice Question: Integration by Substitution
Find
∫2x32x2+1dx\displaystyle\int_{ }^{ }\frac{2x^3}{2x^2+1}dx
Practice Question: Integration by Substitution
Find
∫cot⁡x ln⁡(sin⁡x)dx\displaystyle\int\cot x\ \ln\left(\sin x\right)dx
Evaluate the integral ∫1arcsin⁡x1−x2dx\displaystyle\int_{ }^{ }\frac{1}{\arcsin x\sqrt{1-x^2}}dx. (If you need to input an inverse trigonometric function, the computer asks that you use the "arc" notation instead of "-1", i.e. arcsin⁡\arcsin instead of sin⁡−1\sin^{-1}.)
Find the indefinite integral ∫x1+x2dx\int\frac{x}{1+x^2}dx.
Evaluate the integral ∫x3(1+x4)14dx\displaystyle \int_{ }^{ }x^3(1+x^4)^{\frac{1}{4}}dx.
Find the definite integral ∫14exxdx\displaystyle\int_1^4\frac{e^{\sqrt{x}}}{\sqrt{x}}dx.
Special u-substitution
Evaluate ∫xx−53  ⁣dx\int\frac{x}{\sqrt[3]{x-5}}\de{x}.
Combination of techniques: Integration by substitution
Determine ∫1(1+x)2  ⁣dx\int\frac{1}{(1+\sqrt{x})^2}\de{x}.
Evaluate the integral ∫1ecos⁡(ln⁡x)xdx\displaystyle\int_1^e\frac{\cos\left(\ln x\right)}{x} \text{d}x.
Evaluate the integral ∫1arcsin⁡x1−x2dx\displaystyle\int_{ }^{ }\frac{1}{\arcsin x\sqrt{1-x^2}}dx.
Find the value of the definite integral ∫14exxdx\displaystyle \int_1^4\frac{e^{\sqrt{x}}}{\sqrt{x}}dx.
Evaluate the integral ∫1ecos⁡(ln⁡x)xdx\displaystyle\int_1^e\frac{\cos\left(\ln x\right)}{x}dx.
Evaluate the integral ∫sin⁡2x1+sin⁡2xdx\displaystyle\int_{ }^{ }\frac{\sin2x}{1+\sin^2x}dx.
Evaluate the integral ∫cos⁡x sec⁡2(sin⁡x)dx\displaystyle\int_{ }^{ }\cos x\ \sec^2\left(\sin x\right)dx.
Find ∫2x32x2+1dx\displaystyle\int_{ }^{ }\frac{2x^3}{2x^2+1}dx
Find ∫3x(x2−6)dx\displaystyle\int_{ }^{ }\frac{3x}{\sqrt{\left(x^2-6\right)}}dx.
Evaluate ∫−102x23−3x3dx\int_{-1}^0\frac{2x^2}{3-3x^3}dx.
Find ∫12exxdx\int_{ }^{ }\frac{1}{2e^{\sqrt{x}}\sqrt{x}}dx
Integration by Substitution
Find the u expression needed to integrate ∫sec⁡2x tan⁡2x dx\int_{ }^{ }\sec^2x\ \tan^2x\ dx.
Integration by Substitution
Find the u expression needed to integrate ∫sec⁡2xtan⁡x dx\int_{ }^{ }\sec^2x\tan x\ dx.
Integration by Substitution
Find ∫5x2(x3−1)6dx\int_{ }^{ }5x^2\left(x^3-1\right)^6dx.
Integration by Substitution
Evaluate the integral ∫1e2ln⁡xxdx=\int_1^e\frac{2\ln x}{x}dx=
Integration by Substitution
If ∫25f(2x) dx=6\int_2^5f\left(2x\right)\ dx=6 and ∫210f(x)=20\int_2^{10}f\left(x\right)=20, find ∫24f(x) dx\int_2^4f\left(x\right)\ dx.
Integration by Substitution
∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx
Practice: Property of Definite Integral
Practice: Property of Definite Integral
Suppose f(x)f(x) is continuous and let ∫07f(x)dx=7\displaystyle \int_0^7f\left(x\right)dx=7 and ∫710f(x)dx=10\displaystyle \int_7^{10}f\left(x\right)dx=10.
Find 5∫02f(5x)dx\displaystyle 5\int_0^2f\left(5x\right)dx.
Practice: Integration by Substitution (with back-substitution)
Practice: Integration by Substitution (with back-substitution)
Find ∫2x32x2+1dx\int_{ }^{ }\frac{2x^3}{2x^2+1}dx.
Practice: Integration by Substitution
Evaluate ∫02(x−2)e(x2−4x)dx\int_0^2\left(x-2\right)e^{\left(x^2-4x\right)}dx.
Practice: Integration by Substitution (~F2018 Final Q36)
Practice: Integration by Substitution
Find ∫cot⁡x ln⁡(sin⁡x)dx\int_{ }^{ }\cot x\ \ln\left(\sin x\right)dx.
Practice: Integration by Substitution (~F2017 Final Q35)
Practice: Integration by Substitution
Evaluate ∫1e(ln⁡x−5)2xdx\int_1^e\frac{\left(\ln x-5\right)^2}{x}dx.
Practice: Property of Definite Integral
Practice: Property of Definite Integral
Suppose f(x) is continuous and let ∫07f(x)dx=7\int_0^7f\left(x\right)dx=7 and ∫710f(x)dx=10\int_7^{10}f\left(x\right)dx=10. Find 5∫02f(5x)dx5\int_0^2f\left(5x\right)dx
Evaluate ∫01x31+x2 dx\displaystyle \int_{0}^{1} \frac{x^3}{\sqrt{1+x^2}} \,dx
More Properties of Definite Integrals Questions:
Properties of definite integrals
If ∫08f(x)dx=5,∫03f(x)dx=4,\int_0^8f(x)dx=5,\int_0^3f(x)dx=4, and ∫48f(x)dx=−2\int_4^8f(x)dx=-2, then what is the value of
∫34f(x)dx\int_3^4f(x)dx?
Evaluate the definite integral ∫01x(5−x2)dx\int_0^1x(5-x^2)dx.
Evaluate the definite integral ∫01x(5−x2)dx\int_0^1x(5-x^2)dx.
Evaluate the definite integral ∫01x(5−x2)dx\int_0^1x(5-x^2)dx.
Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
Definite with Absolute Value
Evaluate the following integral: ∫03∣x2−x−2∣ dx\displaystyle\int_0^3|x^2-x-2| \ dx
Definite Integral: Trigonometric Function and Absolute Value
Evaluate the definite integral: ∫05π/4∣sin⁡x∣ dx\displaystyle\int_0^{5\pi/4} |\sin x| \ dx
Practice: Definite with Fraction
Q:\textbf{Q:} Evaluate ∫14[1x2−x]dx\displaystyle \int_1^4\left[\frac{1}{x^2}-x\right]dx
Practice: Definite with Absolue Value
Q.\textbf{Q.} Compute the definite integral ∫−22(2−∣x∣)dx\displaystyle \int_{-2}^{2}(2-\lvert x\rvert)\text{d}x
Properties of Definite Integrals
Suppose that f(−x)=f(x)f\left(-x\right)=f\left(x\right) and g(−x)=−g(x)g\left(-x\right)=-g\left(x\right). If ∫−102f(x)=5\int_{-1}^02f\left(x\right)=5, the value of ∫−11[3f(x)+x⋅sin⁡x⋅g(x)]dx=\int_{-1}^1\left[3f\left(x\right)+x\cdot\sin x\cdot g\left(x\right)\right]dx=
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx
Find ∫0π∣cos⁡ x∣dx\int_0^{\pi}\left|\cos\ x\right|dx
Practice: Definite with Absolue Value
Q.\textbf{Q.} Compute the definite integral ∫−22(2−∣x∣)dx\displaystyle \int_{-2}^{2}(2-\lvert x\rvert)\text{d}x
Definite with Absolute Value
Evaluate the following integral: ∫03∣x2−x−2∣ dx\displaystyle\int_0^3|x^2-x-2| \ dx
Definite Integral: Trigonometric Function and Absolute Value
Evaluate the definite integral: ∫05π/4∣sin⁡x∣ dx\displaystyle\int_0^{5\pi/4} |\sin x| \ dx
Practice: Definite with Fraction
Q:\textbf{Q:} Evaluate ∫14[1x2−x]dx\displaystyle \int_1^4\left[\frac{1}{x^2}-x\right]dx
Properties of definite integrals
If ∫08f(x)dx=5,∫03f(x)dx=4,\int_0^8f(x)dx=5,\int_0^3f(x)dx=4, and ∫48f(x)dx=−2\int_4^8f(x)dx=-2, then what is the value of
∫34f(x)dx\int_3^4f(x)dx?
Find the value of the definite integral ∫02x(x+1)dx\displaystyle \int_0^2x(x+1)dx.
Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
Evaluate the definite integral ∫01x(5−x2)dx\int_0^1x(5-x^2)dx.
Antiderivatives: Trigonometric Functions
Evaluate ∫π3π∣cos⁡x∣dx\int_{\frac{\pi}{3}}^{\pi}\left|\cos x\right|dx.