Question 1

Which of the following is the correct antiderivative of $f(x) = x^3$?

Accepted Answer

$\frac{1}{4}x^4 + C$

Question 2

A medication's concentration in the bloodstream is modeled by $f(t) = e^t$. What is the indefinite integral $\int e^t dt$ representing the total exposure over time?

Accepted Answer

$e^t + C$

Question 3

Find the indefinite integral $\int \sin(x) dx$.

Accepted Answer

$-\cos(x) + C$

Question 4

Which of the following is the correct antiderivative of $f(x) = \csc^2(x)$?

Accepted Answer

$-\cot(x) + C$

Question 5

A physicist models the velocity of a particle as $v(t) = 3t^2 + t - 7$. What is the position function $s(t)$, given $s(0) = 0$?

Accepted Answer

$t^3 + \frac{1}{2}t^2 - 7t$

Question 6

Evaluate the indefinite integral $\int 6x(x^2 + 2) dx$ using substitution.

Accepted Answer

$3x^4 + 12x^2 + C$

Question 7

Find the indefinite integral $\int x^2 \sqrt{1 + x^2} dx$ using substitution.

Accepted Answer

$\frac{1}{3}(1 + x^2)^{3/2} - \sqrt{1 + x^2} + C$

Question 8

A biologist is studying a population modeled by $f(x) = \sin(3x)$. What is the indefinite integral $\int \sin(3x) dx$?

Accepted Answer

$-\frac{1}{3}\cos(3x) + C$

Question 9

Find the indefinite integral $\int x^2 e^{x^2} dx$ using substitution.

Accepted Answer

$\frac{1}{2} e^{x^2} + C$

Question 10

Evaluate the definite integral $\int_0^2 x^2 \sqrt{1 + x^2} dx$ using substitution.

Accepted Answer

$\left[ \frac{1}{3}(1 + x^2)^{3/2} - \sqrt{1 + x^2} \right]_0^2$

Question 11

Find the indefinite integral $\int \sqrt{2x - 1} dx$ using substitution.

Accepted Answer

$\frac{2}{3}(2x - 1)^{3/2} + C$

Question 12

A chemical reaction rate is modeled by $f(x) = \sin^2(3x)\cos(3x)$. What is the indefinite integral $\int \sin^2(3x)\cos(3x) dx$?

Accepted Answer

$\frac{1}{9}\sin^3(3x) + C$

Question 13

Find the indefinite integral $\int \frac{3 + x}{1 + x^2} dx$ using split fraction and substitution.

Question 14

Evaluate the definite integral $\int_{-2}^2 \frac{1}{(x-4)^2} dx$.

Accepted Answer

$\left[ -\frac{1}{x-4} \right]_{-2}^2$

Question 15

Find the indefinite integral $\int x \sqrt{x^2 - 1} dx$ using substitution.

Accepted Answer

$\frac{1}{3}(x^2 - 1)^{3/2} + C$

Question 16

Which of the following is the correct antiderivative of $f(x) = 	an(x)$?

Accepted Answer

$-\ln|\cos(x)| + C$

Question 17

A population grows at a rate $f(t) = \sec^2(t)$. What is the total population after time $t$?

Accepted Answer

$	an(t) + C$

Question 18

Find the indefinite integral $\int x^2 e^{x^2} dx$ using substitution, as it might appear in a physics context for a Gaussian distribution.

Accepted Answer

$\frac{1}{2} e^{x^2} + C$

Integration Review: Antiderivatives and Substitution Techniques

Study Guide - Practice Questions

Study Guide - Flashcards