Question 1

Which of the following is the correct formula for integration by parts?

Accepted Answer

$$\int u \, dv = u v - \int v \, du$$

Question 2

Given $$\int x e^x \, dx$$, which functions should you choose for $$u$$ and $$dv$$ when using integration by parts?

Accepted Answer

$$u = x$$, $$dv = e^x \, dx$$

Question 3

Evaluate $$\int \ln x \, dx$$ using integration by parts.

Accepted Answer

$$x \ln x - x + C$$

Question 4

Which substitution is appropriate for evaluating $$\int \cos(5x) \, dx$$?

Accepted Answer

$$u = 5x$$, $$du = 5 \, dx$$

Question 5

What is the result of $$\int x^2$ e^x \, dx$$ using integration by parts twice?

Accepted Answer

$$x^2 e^x - 2x e^x + 2 e^x + C$

Question 6

If $$u = \sin x$$ and $$dv = e^x \, dx$$, what is $$\int \sin x \, e^x \, dx$$ using integration by parts?

Accepted Answer

$$e^x \sin x - \int e^x \cos x \, dx$$

Question 7

Which of the following is the correct substitution for $$\int \frac{1}{x^2$+1} \, dx$$?

Accepted Answer

$$x = 	an 	heta$$

Question 8

A medical researcher models the rate of drug absorption as $$A(t) = t e$$^{-t}$$. What is the total amount absorbed from t=0$ to t$$=\infty$$?

Accepted Answer

$$1$$

Question 9

Which step is NOT part of the substitution rule for integration?

Accepted Answer

Multiply by the derivative of $$u$$ after integrating

Question 10

Evaluate $$\int x \cos x \, dx$$ using integration by parts.

Accepted Answer

$$x \sin x - \int \sin x \, dx$$

Question 11

Which of the following is the correct result for $$\int e^x \sin x \, dx$$?

Accepted Answer

$$\frac{e^x}{2}(\sin x - \cos x) + C$$

Question 12

A physicist needs to compute $$\$$int_0$$^1 x^3$ e$$^{x^2}$$ \, dx$$ for a probability density. What substitution should be used?

Accepted Answer

$$u = x^2$

Question 13

Which of the following is the correct result for $$\int x \ln x \, dx$$?

Accepted Answer

$$\frac{x^2$}{2} \ln x - \frac{x^2$}{4} + C$$

Question 14

Which of the following is a valid application of integration by parts in economics for finding consumer surplus?

Accepted Answer

Integrating the product of price and demand functions

Question 15

If $$u = \ln x$$ and $$dv = x^2$ \, dx$$, what is $$\int x^2$ \ln x \, dx$$ using integration by parts?

Accepted Answer

$$\frac{x^3$}{3} \ln x - \frac{x^3$}{9} + C$$

Question 16

Which of the following is the correct result for $$\$$int_0$$^\pi x \sin x \, dx$$?

Accepted Answer

$$\pi$$

Question 17

A chemist models the concentration of a reactant as $$C(t) = t^2$ e$$^{-t}$$ over t$$ \geq 0$$. What is $$\$$int_0$$^\infty $$t^2$$ $$e^{-t}$$ \, dt$$?

Accepted Answer

$$2$$

Question 18

Which of the following is the correct result for $$\int e$$^{2x}$$ \cos x \, dx$$?

Accepted Answer

$$\frac{e$$^{2x}$$}{5}(2 \cos x + \sin x) + C$$

Question 19

Which of the following is the correct result for $$\int x^3$ \cos x \, dx$$ using integration by parts?

Accepted Answer

$$x^3$$ \sin x - $$3x^2$$ \cos x - 6x \sin x + 6 \cos x + C$$

Integration Techniques: Substitution and Integration by Parts

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