Question 1

Which of the following best describes the integration by parts formula?

Accepted Answer

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

Question 2

Given $$\int x e^x \, dx$$, which substitution is most appropriate for integration by parts?

Accepted Answer

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

Question 3

What is the result of $$\int x \cos x \, dx$$ using integration by parts?

Accepted Answer

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

Question 4

Which of the following is a key insight about integration by parts?

Accepted Answer

It brings a slight change to the integrand which may bring a big change to the integral.

Question 5

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

Accepted Answer

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

Question 6

Which of the following is the reduction formula for $$\int \cos^n x \, dx$$?

Accepted Answer

$$\int \cos^n x \, dx = \frac{1}{n} \sin x \cos$$^{n-1}$$ x + \frac{n-1}{n} \int \cos$$^{n-2}$$ x \, dx$$

Question 7

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

Accepted Answer

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

Question 8

Which of the following is a special trick when integrating $$\int e^x \cos x \, dx$$?

Accepted Answer

Use integration by parts twice and solve for the original integral.

Question 9

What is the result of $$\int e^x \cos x \, dx$$ after applying integration by parts twice?

Accepted Answer

$$\frac{1}{2} e^x (\sin x + \cos x) + C$$

Question 10

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$$\frac{d}{dx}(f(x)g(x)) = f(x)g'(x) + f'(x)g(x)$$

Question 11

Suppose you are evaluating $$\int x \sin x \, dx$$ using integration by parts. What should you choose for $$u$$ and $$dv$$?

Accepted Answer

$$u = x$$, $$dv = \sin x \, dx$$

Question 12

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

Accepted Answer

$$-x \cos x + \int \cos x \, dx$$

Question 13

Which of the following is a valid strategy for choosing $$u in $$integration by parts?

Accepted Answer

Choose $$u so $$that its derivative $$du is $$simpler than $$u.$$

Question 14

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

Accepted Answer

$$\frac{1}{2} x^2$ \ln x - \frac{1}{4} x^2$ + C$$

Question 15

Which of the following is the correct integration by parts formula for definite integrals from $$a to$$ $$b$$?

Accepted Answer

$$\int_a^b u \, dv = [uv]_a^b - \int_a^b v \, du$$

8.1 Integration by Parts

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