Question 1

Which substitution is most appropriate to evaluate the integral $$\int \frac{dx}{x^2$ + 2x + 2}$$?

Accepted Answer

Let $$x + 1 = 	an 	heta$$

Question 2

What is the result of completing the square for the denominator $$x^2 + 2x + 2$?

Accepted Answer

$$(x + 1)^2$ + 1$$

Question 3

Evaluate $$\$$int_0$$^1 \frac{dx}{x^2$ + 2x + 2}$$ using an appropriate trigonometric substitution.

Accepted Answer

$$\frac{\pi}{4}$$

Question 4

Which integration technique is used to solve $$\int x \ln(x) dx$$?

Accepted Answer

Integration by parts

Question 5

Given $$u = \ln(x)$$ and $$dv = x dx$$, what is $$du in $$integration by parts?

Accepted Answer

$$\frac{1}{x} dx$$

Question 6

Evaluate $$\$$int_0$$^1 x \ln(x) dx.$$

Accepted Answer

$$-\frac{1}{4}$$

Question 7

Which of the following is the correct result for $$\int \cot(	heta) d	heta$$?

Accepted Answer

$$\ln|\sin(	heta)| + C$$

Question 8

Evaluate $$\$$int_0$$^1 (x^2$ - x) dx.$$

Accepted Answer

$$-\frac{1}{6}$$

Question 9

Which method is used to solve $$\int \frac{dx}{x(x-2)}$$?

Accepted Answer

Partial fractions

Question 10

What is the partial fraction decomposition of $$\frac{1}{x(x-2)}$$?

Accepted Answer

$$\frac{1}{2} \left( \frac{1}{x} - \frac{1}{x-2} \right)$$

Question 11

Evaluate $$\$$int_0$$^2 \frac{dx}{x(x-2)}.$$

Accepted Answer

$$0$$

Question 12

What is the general solution to $$\int \frac{dx}{x^2$ + 2x + 2}$$?

Accepted Answer

$$\arctan(x+1) + C$$

Question 13

Evaluate $$\int_{-\infty}^{\infty} \frac{dx}{x^2$ + 2x + 2}.$$

Accepted Answer

$$2\pi$$

Question 14

Which of the following is the correct answer for $$\int x^2$ e$$^{ax}$$ dx$$?

Accepted Answer

$$\frac{e$$^{ax}$$}{a} (x^2$ - \frac{2x}{a} + \frac{2}{a^2$}) + C$$

Question 15

If $$\int x^2$ e$$^{ax}$$ dx is $$evaluated from $$x = -\frac{1}{a} to$$ $$x = \frac{2}{a}$$, what is the result?

Accepted Answer

$$\frac{2e^2$}{a} - \frac{2e$$^{-1}$$}{a}$$

Question 16

Which substitution is used to evaluate $$\int \frac{dx}{x^2$ + 2x + 2} in $$terms of $$	heta$$?

Accepted Answer

$$x + 1 = 	an 	heta$$

Question 17

What is the value of $$\$$int_0$$^1 x \ln(x) dx in $$terms of definite integration?

Accepted Answer

$$-\frac{1}{4}$$

Question 18

Which of the following is the correct answer for $$\int \cot(	heta) d	heta$$?

Accepted Answer

$$\ln|\sin(	heta)| + C$$

Integration Techniques and Applications: Calculus Study Notes

Study Guide - Practice Questions