Question 1

Which of the following is the correct statement of the substitution rule for definite integrals?

Accepted Answer

$$\int_{a}^{b} f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$$

Question 2

Given $$\int \cos(x)\,dx$$, which of the following is the correct antiderivative?

Accepted Answer

$$\sin(x) + C$$

Question 3

Evaluate $$\int_{0}^{\pi/2} \sin(x)\,dx$$ using the substitution rule.

Accepted Answer

$$1$$

Question 4

Which substitution is appropriate for evaluating $$\int x e$$^{x^2}$$ dx$$?

Accepted Answer

$$u = x^2$

Question 5

What is the result of $$\int \frac{1}{x} dx$$?

Accepted Answer

$$\ln|x| + C$$

Question 6

Find $$\int \sec^2$(x) dx.$$

Accepted Answer

$$	an(x) + C$$

Question 7

Evaluate $$\int_{1}^{e} \frac{1}{x} dx.$$

Accepted Answer

$$1$$

Question 8

Which of the following is the correct antiderivative of $$\int e$$^{x}$$ dx$$?

Accepted Answer

$$e^{x} + C$

Question 9

If $$u = 2x + 1$$, what is $$du in $$terms of $$dx$$?

Accepted Answer

$$2 dx$$

Question 10

Evaluate $$\int \frac{1}{\sqrt{1-x^2$}} dx.$$

Accepted Answer

$$\arcsin(x) + C$$

Question 11

Which substitution would simplify $$\int \sin(3x+2) dx$$?

Accepted Answer

$$u = 3x + 2$$

Question 12

Find $$\int \frac{1}{1+x^2$} dx.$$

Accepted Answer

$$\arctan(x) + C$$

Question 13

Evaluate $$\int_{0}^{1} 2x e$$^{x^2}$$ dx$$ using substitution.

Accepted Answer

$$e^{1}$$ - $$e^{0}$$

Question 14

Which of the following is the correct formula for $$\int 	an(x) dx$$?

Accepted Answer

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

Question 15

Find $$\int \frac{1}{\sqrt{x^2$-1}} dx.$$

Accepted Answer

$$\arcsec(x) + C$$

Question 16

Evaluate $$\int_{0}^{\pi} \sin(x) dx.$$

Accepted Answer

$$0$$

Question 17

Which of the following is the correct formula for $$\int \sec(x) dx$$?

Accepted Answer

$$\ln|\sec(x) + 	an(x)| + C$$

Question 18

If $$u = x^2$, what is du$?

Accepted Answer

$$2x dx$$

Question 19

Find $$\int e$$^{2x}$$ dx.$$

Accepted Answer

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

Question 20

Evaluate $$\int \frac{1}{x^2$+1} dx.$$

Accepted Answer

$$\arctan(x) + C$$

Integration Techniques: Substitution Rule and General Integration Formulas

Study Guide - Practice Questions