Q: For the integral $$\int \sec^n(x) \tan^m(x) \, dx$$, which identity is useful for reducing the power of tangent?

$$\tan^2$(x) + 1 = \sec^2$(x)$$

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

Given $$\int x^2$ e$$^{3x}$$ \, dx$$, which function should you choose for $$u$$ when applying integration by parts?

Accepted Answer

$$u = x^2$

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 a valid choice for $$dv$$ when integrating $$\int x^2$ \ln(x) \, dx by $$parts?

Accepted Answer

$$dv = x^2$ \, dx$$

Question 5

What is the result of $$\int x^2$ \ln(x) \, dx$$ using integration by parts?

Accepted Answer

$$\frac{x^3$}{3} \ln(x) - \frac{x^3$}{9} + C$$

Question 6

When integrating $$\int e$$^{4x}$$ \cos(5x) \, dx$$, what is the general strategy?

Accepted Answer

Apply integration by parts twice and solve for the original integral.

Question 7

What is the value of $$\$$int_0$$^{\frac{\pi}{2}} \cos^3$(x) \, dx$$?

Accepted Answer

$$\frac{2}{3}$$

Question 8

Which reduction formula is used for $$\int \sin^m(x) \cos^n(x) \, dx$$ when both $$m$$ and $$n$$ are even?

Accepted Answer

Power-reduction identities

Question 9

Find the volume of the solid formed by rotating the region bounded by $$f(x) = \cos(x)$$, $$0 \leq x \leq \frac{\pi}{2}$$, and $$y = 0$$ about the y-axis.

Accepted Answer

$$2\pi \left( \frac{\pi}{2} - 1 \right)$$ cubic units

Question 10

Which of the following is the correct reduction identity for $$\cos^2$(x)$$?

Accepted Answer

$$\cos^2$(x) = \frac{1}{2} + \frac{1}{2} \cos(2x)$$

Question 11

Evaluate $$\int e$$^{4x}$$ \cos(5x) \, dx.$$

Accepted Answer

$$\frac{1}{41} e$$^{4x}$$ (4 \cos(5x) + 5 \sin(5x)) + C$$

Question 12

Which of the following is a correct application of the product rule for differentiation?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Question 13

For the integral $$\int \sec^n(x) 	an^m(x) \, dx$$, which identity is useful for reducing the power of tangent?

Accepted Answer

$$	an^2$(x) + 1 = \sec^2$(x)$$

Question 14

Which of the following best describes the 'good neighbor' strategy for integrating $$\int \sin^m(x) \cos^n(x) \, dx$$ when one power is odd?

Accepted Answer

Save one factor of the odd power and convert the rest using the Pythagorean identity.

Question 15

Evaluate $$\$$int_0$$^{\frac{\pi}{4}} \sin^3$(x) \, dx.$$

Accepted Answer

$$1 - \frac{1}{3} \left( \frac{\sqrt{2}}{2} \$$right)^3$$

Question 16

Which of the following is the correct derivative of $$	an(x)$$?

Accepted Answer

$$\frac{d}{dx} 	an(x) = \sec^2$(x)$$

Integration Techniques: Integration by Parts and Trigonometric Integrals

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