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

Which of the following is NOT a condition for using integration by parts?

Accepted Answer

The integrand must be a rational function.

Question 3

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

Accepted Answer

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

Question 4

In the integral $$\int x^2$ \ln x \, dx$$, which function should you choose as $$u$$ for integration by parts?

Accepted Answer

$$\ln x$$

Question 5

Find $$\int \ln x \, dx$$ for $$x \u003e 0$$ using integration by parts.

Accepted Answer

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

Question 6

Which of the following real-world scenarios best illustrates the use of integration by parts?

Accepted Answer

Calculating the work done by a variable force $$F(x) = x e^x$$ over a distance.

Question 7

Evaluate $$\int (2x + 5) e$$^{-x}$$ \, dx$$ using integration by parts.

Accepted Answer

$$-(2x + 7) e$$^{-x}$$ + C$$

Question 8

If $$u = x^2$ and dv$$ = \ln x \, dx$$, what is du$?

Accepted Answer

$$2x \, dx$$

Question 9

Which step is necessary when applying integration by parts twice to $$\int x^2$ e^x \, dx$$?

Accepted Answer

After the first application, set $$u = x$$ and $$dv = e^x \, dx$$ for the remaining integral.

Question 10

Which of the following integrals is most suitable for the integration by parts method?

Accepted Answer

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

Question 11

A pharmaceutical company models the rate of drug absorption as $$A(t) = t e$$^{-t}$$. Which integral would you use to find the total absorption over time using integration by parts?

Accepted Answer

$$\int t e$$^{-t}$$ \, dt$$

Question 12

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

Accepted Answer

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

Question 13

Which of the following is a common mistake when applying integration by parts?

Accepted Answer

Choosing $$dv as a $$function that is harder to integrate than the original integrand.

Question 14

Evaluate $$\int x^2$ e^x \, dx$$ using integration by parts twice.

Accepted Answer

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

Question 15

Which of the following is the best choice for $$u in $$integration by parts for $$\int x \ln x \, dx$$?

Accepted Answer

$$\ln x$$

Question 16

A physicist needs to compute $$\int t \sin t \, dt to $$find the displacement of a particle under a sinusoidal force. What is the first step in applying integration by parts?

Accepted Answer

Let $$u = t$$ and $$dv = \sin t \, dt$$

Integration by Parts (Section 7.1): Techniques and Examples

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