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.

Accepted Answer

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

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