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Find the definite integral ∫_0^2x(x+1)dx.

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Wize University Calculus 1 Textbook > Integrals

The Fundamental Theorem of Calculus - Part 2

6 Activities

Wize University Calculus 1 Textbook > Integrals

Properties of Definite Integrals

3 Activities

Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
More The Fundamental Theorem of Calculus - Part 2 Questions:
Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
Practice: Property of Definite Integral
Practice: Property of Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10 and ∫−31f′(x)dx=12\displaystyle \int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
Fundamental Theorem of Calculus
Given that both f(x)f(x) and f′(x)f'(x) are continuous everywhere, and given f(1)=10f(1)=10 and ∫−31f′(x)dx=12\displaystyle\int_{-3}^1f'\left(x\right)dx=12 , find f(−3)f(-3).
Definite Integrals with Trig
Compute: ∫25(x3+sin⁡x)dx\displaystyle \int_{2}^{5}(x^3+\sin{x})\text{d}x
Integrals
Evaluate ∫−30∣(x−1)(x+2)∣dx\int_{-3}^0\left|\left(x-1\right)\left(x+2\right)\right|dx
Trigonometric Functions
If f(x)=arcsin⁡(ln⁡x)f\left(x\right)=\arcsin\left(\ln\sqrt{x}\right), then ∫1ef′(x)dx=\int_1^ef'\left(x\right)dx=
Riemann Sums
Evaluate lim⁡n→∞ ∑i=1n[5n[(1+5in)2−3]]\displaystyle \lim_{n\to\infty}\ \sum_{i=1}^n\left[\frac{5}{n}\left[\left(1+\frac{5i}{n}\right)^2-3\right]\right]
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Practice: Property of Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10and ∫−31f′(x)dx=12\int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
A factory produces pollution at a rate of(t+1) gm/hr(t+1)\ gm/hr. How much does it produce after 4 hours?
Integrals
Evaluate ∫13x3+xxdx\int_1^3\frac{\sqrt{x^3}+x}{x}dx
If ∫02[3(kx)2+(k+1)x+1]dx=10\int_0^2[3(kx)^2+(k+1)x+1]dx=10 then what is a possible value of kk ?
Practice: Property of Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10and ∫−31f′(x)dx=12\int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
Given that f(7)=1  and  f(5)=8f\left(7\right)=1\ \ and\ \ f\left(5\right)=8
find ∫57f′(x) dx\displaystyle \int_{5}^{7} f'(x)\space dx
Derivatives
Evaluate
∫0π(3x2+sin⁡x) dx\int_0^{\pi} (3x^2 + \sin x ) \ dx
Derivatives
Find
∫01(x4−2x3) dx\int_{0}^1(x^4-2x^3)\ dx
Antiderivatives: Trigonometric Functions
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx
Practice: Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10and ∫−31f′(x)dx=12\displaystyle \int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
Practice: Definite Integral
Evaluate ∫0319+x2+2xln⁡2 dx\displaystyle \int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx.
🌶️ Evaluate the limit by recognizing the Riemann sum:
lim⁡n→∞∑i=1n27i3n4.\lim\limits_{n\rightarrow\infin}\displaystyle\sum_{i=1}^n\frac{27i^3}{n^4}.
Enter your answer as a fraction.
Evaluate the integral using the Fundamental Theorem of Calculus: ∫01e2x+e−2xdx\displaystyle\int_0^1e^{2x}+e^{-2x}dx.
Evaluate the integral using the Fundamental Theorem of Calculus ∫242−416−x2dx\displaystyle\int_2^{\frac{4}{\sqrt{2}}}-\frac{4}{\sqrt{16-x^2}}dx.
final114
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Find the exact value of
∫0πsin⁡xdx\int_0^{\pi}\sin xdx
Definite Integrals with Trig
Compute: ∫25(x3+sin⁡x)dx\displaystyle \int_{2}^{5}(x^3+\sin{x})\text{d}x
Fundamental Theorem of Calculus
Given that both f(x)f(x) and f′(x)f'(x) are continuous everywhere, and given f(1)=10f(1)=10 and ∫−31f′(x)dx=12\displaystyle\int_{-3}^1f'\left(x\right)dx=12 , find f(−3)f(-3).
final114
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Evaluate the integral using the Fundamental Theorem of Calculus ∫242−416−x2dx\displaystyle\int_2^{\frac{4}{\sqrt{2}}}-\frac{4}{\sqrt{16-x^2}}dx.
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Practice: Property of Definite Integral
Practice: Property of Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10 and ∫−31f′(x)dx=12\displaystyle \int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
Practice: Property of Definite Integral
Practice: Property of Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10and ∫−31f′(x)dx=12\int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
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
Definite Integrals: Integration by Substitution
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=
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.
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.