High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 1 Textbook > Integrals

The Fundamental Theorem of Calculus - Part 2

The Fundamental Theorem of Calculus - Part 1Previous Section
Integration by Substitution (U-Substitution)Next Section
Popular Courses
Find My Course
0:00 / 0:00

Fundamental Theorem of Calculus (FTC) - Part 2

Computing definite integrals by definition is often difficult and time consuming. Fortunately, the second part of The Fundamental Theorem of Calculus provides us with a straightforward way of computing definite integrals.

Fundamental Theorem of Calculus: 2

Let f(x)f(x) be a continuous function on [a,b][a,b], so that F′(x)=f(x)F'(x)=f(x) on [a,b][a,b]we have
∫abf(x)dx=F(b)−F(a)\boxed{\int^b_af(x)dx=F(b)-F(a)}

Note: Sometimes we will use the notation F(x)∣ab=F(b)−F(a)F(x) \bigg|_a^b=F(b)-F(a)

Wize Concept
To compute definite integrals using FTC, compute the antiderivative of the function, substitute the upper bound, and subtract substituting the lower bound.

0:00 / 0:00

Example: FTC Part 2

Evaluate the integral using the Fundamental Theorem of Calculus.

∫123xdx\displaystyle \int_1^23\sqrt{x}dx

∫123xdx\displaystyle \int_1^23\sqrt{x}dx

=∫123x12dx \displaystyle=\int_1^23x^{\frac{1}{2}}dx\

=3⋅x3232∣12=\displaystyle \left. \displaystyle 3\cdotp \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right|_1^2 (by defining the antiderivative)


=2⋅(2)32−2(1)32=\displaystyle2 \cdotp (2)^{\frac{3}{2}}-2(1)^{\frac{3}{2}} (applying FTC2)

=42−2=4\sqrt{2}-2

0:00 / 0:00

Example: FTC Part 2

Evaluate the integral using the Fundamental Theorem of Calculus.

∫14e−3x+2x−1x12dx\displaystyle \int_1^4e^{-3x}+2^x-\frac{1}{x^{\frac{1}{2}}}dx


∫14e−3x+2x−1x12dx\displaystyle \int_1^4e^{-3x}+2^x-\frac{1}{x^{\frac{1}{2}}}dx

=∫14(e−3x+2x−x−12)dx=\displaystyle \int_1^4 (e^{-3x}+2^x-{x^{-\frac{1}{2}}})dx

=[−13e−3x+1ln⁡22x−(1)x1212]14=\displaystyle \left[-\frac{1}{3}e^{-3x}+\frac{1}{\ln2}2^x-\left(1\right)\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right]_1^4

=[−13e−3(4)+1ln⁡224−(2)412]−[−13e−3(1)+1ln⁡221−(2)112]=−13e12+16ln⁡2−4−[−13e3+2ln⁡2−2]=−13e12+13e3+14ln⁡2−2\begin{aligned} &= \left[ -\frac{1}{3}e^{-3(4)} + \frac{1}{\ln2}2^4 - (2)4^{\frac{1}{2}}\right] - \left[ -\frac{1}{3}e^{-3(1)} + \frac{1}{\ln2}2^1 - (2)1^{\frac{1}{2}}\right]\\ \\ &= \frac{-1}{3e^{12}} + \frac{16}{\ln2}-4-\left[ \frac{-1}{3e^3} + \frac{2}{\ln2} -2\right]\\ \\ &=-\frac{1}{3e^{12}}+\frac{1}{3e^3}+\frac{14}{\ln2}-2 \end{aligned}

Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx.
If ∫02[3(kx)2+(k+1)x+1]dx=10\displaystyle \int_0^2[3(kx)^2+(k+1)x+1]dx=10, then what is a possible value of kk ?
Given that both f(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).

Extra Practice
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?
Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
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).