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

Suppose you want to evaluate $$\int x e^x dx$$ using integration by parts. Which choice of $$u$$ and $$dv is $$most appropriate?

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

Both functions must be polynomials.

Question 5

Find $$\int \ln x dx$$ using integration by parts.

Accepted Answer

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

Question 6

A pharmaceutical company needs to calculate $$\$$int_0$$^1 x e^x dx to $$determine the expected value of a drug's concentration over time. What is the value?

Accepted Answer

$$e - 2$$

Question 7

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 8

Which of the following integrals is best solved using integration by parts?

Accepted Answer

$$\int x \ln x dx$$

Question 9

Find $$\int x \ln x dx$$ using integration by parts.

Accepted Answer

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

Question 10

A physicist is modeling the work done by a force $$F(x) = (2x + 5) e$$^{-x}$$ over the interval $$[0, 1]$$. What is $$\$$int_0^1 (2x + 5$$) $$e^{-x} dx$?

Accepted Answer

$$-2 e$$^{-1}$$ + 2 + 5 (1 - e$$^{-1}$$)$$

Question 11

Which of the following is the correct first step in applying integration by parts to $$\int x^2$ \ln x dx$$?

Accepted Answer

Let $$u = \ln x$$, $$dv = x^2$ dx$$

Question 12

Evaluate $$\$$int_1$$^e \ln x dx.$$

Accepted Answer

$$e - 1$$

Question 13

A biologist models population growth with $$P(t) = t e$$^{2t}$$. What is $$\int t $$e^{2t} dt$?

Accepted Answer

$$\frac{1}{2} t e$$^{2t}$$ - \frac{1}{4} e$$^{2t}$$ + C$$

Question 14

Which of the following is a valid reason to use integration by parts?

Accepted Answer

The integrand is a product of a polynomial and a transcendental function.

Question 15

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

Accepted Answer

$$x \sin x + \cos x + C$$

Question 16

A chemical engineer needs to compute $$\$$int_0$$^1 x e$$^{-x}$$ dx to $$model a reaction rate. What is the value?

Accepted Answer

$$1 - 2 e$$^{-1}$$

Question 17

Evaluate $$\int x^3$ e^x dx$$ using integration by parts three times.

Accepted Answer

$$x^3 e^x - 3$$ $$x^2 e^x + 6 x e^x - 6 e^x + C$

Question 18

Which of the following is the best choice for $$u$$ when integrating $$\int x^2$ \ln x dx by $$parts?

Accepted Answer

$$u = \ln x$$

Question 19

A medical researcher models the decay of a drug with $$C(t) = (2t + 5) e$$^{-t}$$. What is $$\int (2t + 5) $$e^{-t} dt$?

Accepted Answer

$$-2 t e$$^{-t}$$ - (2 + 5) e$$^{-t}$$ + C$$

Integration by Parts (Section 7.1): Techniques and Examples

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