Question 1

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

Accepted Answer

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

Question 2

When using integration by parts for $$\int x e$$^{3x}$$ \, dx$$, which is the best choice for $$u$$?

Accepted Answer

$$u = x$$

Question 3

Evaluate $$\int x \ln x \, dx$$ using integration by parts. What is the correct first step for choosing $$u$$ and $$dv$$?

Accepted Answer

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

Question 4

Which of the following integrals is best suited for integration by parts rather than $$u-$$substitution?

Accepted Answer

$$\int x^2$ \ln x \, dx$$

Question 5

Given $$\int x^2$ e$$^{-x}$$ \, dx$$, what is the best choice for $$u$$ and $$dv$$ according to the summary table?

Accepted Answer

$$u = x^2$, dv$$ = $$e^{-x}$$ \, dx$$

Question 6

Evaluate $$\int_{1}^{8} \ln(7x) \, dx. $$What does this integral represent graphically?

Accepted Answer

The area under the curve $$y = \ln(7x)$$ from $$x = 1 to$$ $$x = 8$$

Question 7

Which of the following is NOT a good rule for selecting $$u in $$integration by parts?

Accepted Answer

Choose $$u so $$that $$\int v \, du is $$more complicated than the original integral.

Question 8

For the integral $$\int x e$$^{-x^2}$$ \, dx$$, which technique is most appropriate?

Accepted Answer

$$u-$$substitution

Question 9

Suppose you are asked to evaluate $$\int \ln x \, dx. $$What is the correct application of integration by parts?

Accepted Answer

Let $$u = \ln x$$, $$dv = dx$$

Question 10

Evaluate $$\int \frac{\ln x}{x} \, dx. $$What is the result?

Accepted Answer

$$\frac{1}{2} (\ln x)^2$ + C$$

Question 11

A pharmaceutical company models the rate of drug absorption in the bloodstream as $$f(x) = x e$$^{-x}$$, where x$ is time in hours. Which integration technique is best to find the total absorption over time?

Accepted Answer

Integration by parts

Question 12

Which of the following is the correct integration by parts setup for $$\int \sqrt{x} \ln x \, dx$$?

Accepted Answer

$$u = \ln x$$, $$dv = \sqrt{x} \, dx$$

Question 13

If $$u = x^n$$ and $$dv = e$$^{ax}$$ dx$$, what is $$du$$?

Accepted Answer

$$du = n x$$^{n-1}$$ dx$$

Question 14

Which of the following is a real-world application of integration by parts?

Accepted Answer

Calculating the work done by a variable force over a distance

Question 15

Evaluate $$\int_{1}^{e} \ln x \, dx$$ using integration by parts.

Accepted Answer

$$e - 1$$

Integration by Parts: Techniques and Applications

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