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

When using integration by parts, which function should you typically choose for $$u$$?

Accepted Answer

The function that becomes simpler when differentiated.

Question 3

Evaluate $$\int 6x e$$^{x}$$ \, dx$$ using integration by parts.

Accepted Answer

$$6x e$$^{x}$$ - 6 e$$^{x}$$ + C$$

Question 4

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

Accepted Answer

$$e^{x}$$

Question 5

Suppose you are asked to evaluate $$\int x \cos(x) \, dx. $$Which of the following is the best choice for $$u$$?

Accepted Answer

$$u = x$$

Question 6

Integration by parts is derived from which differentiation rule?

Accepted Answer

The product rule

Question 7

If $$u = 6x$$ and $$dv = e$$^{x}$$ dx$$, what are $$du$$ and $$v$$?

Accepted Answer

$$du = 6 dx$$, $$v = e$$^{x}$$

Question 8

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

Accepted Answer

$$\int x \ln(x) \, dx$$

Question 9

After applying integration by parts to $$\int 6x e$$^{x}$$ dx$$, which integral remains to be solved?

Accepted Answer

$$\int 6 e$$^{x}$$ dx$$

Question 10

A pharmaceutical company models the rate of a drug's absorption as $$R(t) = t e$$^{-t}$$, where t$ is time in hours. To find the total amount absorbed from t=0$ to t$$=\infty$$, which integration technique is most appropriate?

Accepted Answer

Integration by parts

Question 11

Which of the following is the correct application of integration by parts to $$\int x e$$^{x}$$ dx$$?

Accepted Answer

$$x e$$^{x}$$ - \int e$$^{x}$$ dx$$

Question 12

Which of the following statements about integration by parts is TRUE?

Accepted Answer

It is useful when integrating the product of two functions where substitution does not work.

Question 13

Evaluate $$\int x \sin(x) dx$$ using integration by parts.

Accepted Answer

$$-x \cos(x) + \int \cos(x) dx$$

Question 14

In the integration by parts formula $$\int u dv = u v - \int v du$$, what does $$dv$$ represent?

Accepted Answer

A differential that is easily integrable

Question 15

A business models its profit as $$P(x) = x e$$^{-x}$$, where x$ is the number of units sold. To find the total profit for x$ from 0$ to $$\infty$$, which method should be used?

Accepted Answer

Integration by parts

Question 16

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

Accepted Answer

Choosing $$u as a $$function that does not simplify when differentiated

Question 17

If $$u = \ln(x)$$ and $$dv = x dx$$, what is $$du$$?

Accepted Answer

$$\frac{1}{x} dx$$

Question 18

Which of the following integrals would NOT benefit from integration by parts?

Accepted Answer

$$\int x^2$ dx$$

Question 19

A company wants to find the total cost over time if the cost rate is $$C(t) = t e$$^{-t}$$ for t$$ \geq 0$$. Which integration technique should be used?

Accepted Answer

Integration by parts

Question 20

Which step is necessary after applying integration by parts to $$\int 6x e$$^{x}$$ dx$$?

Accepted Answer

Integrate $$6 e$$^{x}$$

Integration by Parts: Theory, Formula, and Example

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