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Chapter 5: Q 8. (page 451)

Describe strategy for solving the types of integral given in
exercise $\int \cos^{k} x d x, k$ is odd.

Short Answer

Expert verified

Write $\cos^{k} x = \cos x \cos^{k - 1} x$

Step by step solution

01

Step 1. Given information

Integral is $\int \cos^{k} x d x$

02

Step 2. Explanation

Consider the integral $\int \cos^{k} x d x, k$ is odd.

Write $\cos^{k} x = \cos x \cos^{k - 1} x$ then write $\cos^{2} x = \frac{1 + \cos 2 x}{2}$

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Most popular questions from this chapter

Solve the integral: $\int \frac{x^{2}}{e^{x}} d x$

Solve the integral: $\int x \ln (x) d x$

Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.

$(a) \int \frac{4 + x^{2}}{x} d x$ $(b) \int \frac{x}{4 + x^{2}} d x$

role="math" localid="1648759296940" $(c) \int \frac{x^{2}}{4 + x^{2}} d x$ $(d) \int \frac{16 - x^{4}}{4 + x^{2}} d x$

Find three integrals in Exercises 39–74 that can be solved by using a trigonometric substitution of the form $x = a \tan u$ .

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

$\int u^{2} d u$

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