Unit 2 · Test

Unit 2 test: Advanced Integration

14 questions. Answer them all, then submit to see your score. Solutions unlock after you submit.

This test covers substitution and other review techniques, integration by parts, partial fractions with distinct linear factors, and improper integrals.

Question 1

$\displaystyle\int x^2 e^{x^3}\,dx =$

Question 2

Evaluate $\displaystyle\int_0^3 \frac{x}{\sqrt{x^2 + 16}}\,dx$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 3

Evaluate $\displaystyle\int_1^3 \frac{dx}{x^2 - 4x + 5}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 4

$\displaystyle\int x\sin(2x)\,dx =$

Question 5

Evaluate $\displaystyle\int_0^1 x e^{-x}\,dx$ . Give an exact answer.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 6

Find the antiderivative $F(x)$ of $f(x) = \ln x$ that satisfies $F(1) = 2$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 7

The function $g$ has a continuous second derivative, with $g(1) = 1$ , $g(3) = 7$ , $g'(1) = 2$ and $g'(3) = 4$ . Find $\displaystyle\int_1^3 x\,g''(x)\,dx$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 8

Given $\dfrac{7x - 1}{(x + 1)(x - 3)} = \dfrac{A}{x + 1} + \dfrac{B}{x - 3}$ , find $B$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 9

$\displaystyle\int \frac{x + 5}{x^2 + x - 2}\,dx =$

Question 10

Evaluate $\displaystyle\int_2^4 \frac{dx}{x^2 - 1}$ . Give an exact answer.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 11

Evaluate $\displaystyle\int_1^{\infty} \frac{4}{x^3}\,dx$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 12

What is the value of $\displaystyle\int_1^{\infty} \frac{x}{x^2 + 1}\,dx$ ?

Question 13

Evaluate $\displaystyle\int_0^{\infty} x e^{-x^2}\,dx$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 14

Evaluate $\displaystyle\int_0^1 \ln x\,dx$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.