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Chapter 5: Q 36. (page 452)

Solve the following integral.
$\int \sin x \cos^{4} x d x$

Short Answer

Expert verified

Answer is $\frac{- \cos^{5} x}{5} + C$

Step by step solution

01

Step 1. Given information

An integral is $\int \sin x \cos^{4} x d x$

02

Step 2. Explanation

Let $c o s x = u$

$\begin{array}{rcl} \int \sin x \cos^{4} x d x & = & \int - u^{4} d u \\ = & \frac{- u^{5}}{5} + C \\ = & \frac{- \cos^{5} x}{5} + C \end{array}$

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Most popular questions from this chapter

Solve $\int \frac{x^{2}}{1 + x^{2}} d x$ the following two ways:

(a) with the substitution $u = \tan^{- 1} x;$

(b) with the trigonometric substitution x = tan u.

Consider the integral $\int x {(x^{2} - 1)}^{2} d x$ .

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

(c) How must your two answers be related? Use algebra to prove this relationship.

Find three integrals in Exercises 27–70 for which either algebra or u-substitution is a better strategy than integration by parts.

Why don’t we ever have cause to use the trigonometric substitution $x = \cos u$ ?

Explain why, if $x = a \sin u$ , then $\sqrt{a^{2} - x^{2}} = a \cos u$ . Your explanation should include a discussion of domains and absolute values.

See all solutions

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