Q: Which of the following is the correct result for $$\int \tan$$^{-1}$$ x \, dx$$ using integration by parts?

$$x \tan$$^{-1}$$ x - \frac{1}{2} \ln(1 + x^2$) + c$$

Question 1

Evaluate the integral $$\int 8s^2$ e$$^{3s}$$ \, ds$$ using integration by parts.

Accepted Answer

$$\frac{8}{3} s^2$ e$$^{3s}$$ - \frac{16}{9} s e$$^{3s}$$ + \frac{16}{27} e$$^{3s}$$ + c$$

Question 2

Which of the following is the correct result for $$\int 	an$$^{-1}$$ x \, dx$$ using integration by parts?

Accepted Answer

$$x 	an$$^{-1}$$ x - \frac{1}{2} \ln(1 + x^2$) + c$$

Question 3

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = e$$^{7t}$$ \sin(5t). $$What is $$\int e$$^{7t}$$ \sin(5t) \, dt$$?

Accepted Answer

$$\frac{1}{74} \left( 7 e$$^{7t}$$ \sin(5t) - 5 e$$^{7t}$$ \cos(5t) \right) + c$$

Question 4

Evaluate $$\int \sin^2$ x \cos^2$ x \, dx$$ using an appropriate trigonometric identity.

Accepted Answer

$$\frac{1}{8} x - \frac{1}{32} \sin(4x) + c$$

Question 5

Which of the following is the correct antiderivative for $$\int \sin^5$ x \, dx$$?

Accepted Answer

$$-\frac{1}{5} \cos^5$ x + \frac{2}{3} \cos^3$ x - \cos x + c$$

Question 6

Evaluate $$\int \csc^5$ x \cot^3$ x \, dx.$$

Accepted Answer

$$-\frac{1}{7} \csc^7$ x + \frac{1}{5} \csc^5$ x + c$$

Question 7

A physicist is modeling the displacement of a particle with $$f(x) = \sin^2$ x \cos^2$ x. $$What is the average value of $$f(x) on$$ $$[0, \pi]$$?

Accepted Answer

$$\frac{1}{8}$$

Question 8

Which substitution is most effective for evaluating $$\int \sin^5$ x \, dx$$?

Accepted Answer

Let $$u = \cos x$$, $$du = -\sin x \, dx$$

Question 9

Suppose $$\int f(x) g'(x) \, dx is to be $$evaluated by integration by parts. Which formula should be used?

Accepted Answer

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

Question 10

A medical researcher is analyzing a signal modeled by $$e^{7t}$$ \sin(5t)$$. Which integration technique is most appropriate for $$\int $$e^{7t}$$ \sin(5t) \, dt$$?

Accepted Answer

Integration by parts, applied twice

Question 11

Which trigonometric identity is most useful for simplifying $$\sin^2$ x \cos^2$ x$$ before integrating?

Accepted Answer

$$\sin^2$ x \cos^2$ x = \frac{1}{8} (1 - \cos 4x)$$

Question 12

Which of the following best describes the process for integrating $$\int 	an$$^{-1}$$ x \, dx$$?

Accepted Answer

Let $$u = 	an$$^{-1}$$ x$$, $$dv = dx$$, then use integration by parts

Question 13

A student attempts to integrate $$\int \csc^5$ x \cot^3$ x \, dx by $$letting $$u = \csc x. $$What is $$du in $$terms of $$dx$$?

Accepted Answer

$$du = -\csc x \cot x \, dx$$

Question 14

Which of the following integrals would most likely require integration by parts?

Accepted Answer

$$\int x \ln x \, dx$$

Question 15

A chemical reaction's rate is modeled by $$R(t) = \sin^5$ t. $$What is the total change in concentration from $$t = 0 to$$ $$t = \pi$$?

Accepted Answer

$$0$$

Question 16

Which of the following is a common mistake when integrating $$\int \sin^5$ x \, dx$$?

Accepted Answer

Assuming $$\int \sin^5$ x \, dx = -\frac{1}{5} \cos^5$ x + c$$

Question 17

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

Accepted Answer

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

Step-by-Step Guidance for Integration by Parts and Trigonometric Integrals (Calculus)

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