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Chapter 5: Q 37. (page 452)

Solve the following integral.
$\int \cos^{3} x \sin^{4} x d x$

Short Answer

Expert verified

Answer is $\frac{\sin^{5} x}{5} - \frac{\sin^{7} x}{7} + C$

Step by step solution

01

Step 1. Given information

An integral is $\int \cos^{3} x \sin^{4} x d x$

02

Step 2. Explanation

$\int \cos^{3} x \sin^{4} x d x = \int (1 - \sin^{2} x) \cos x \sin^{4} x d x$

Let width="61">sinx=u

$\begin{array}{rcl} \int \cos^{3} x \sin^{4} x d x & = & \int u^{4} (1 - u^{2}) d u \\ = & \int u^{4} - u^{6} d u \\ = & \frac{u^{5}}{5} - \frac{u^{7}}{7} + C \\ = & \frac{\sin^{5} x}{5} - \frac{\sin^{7} x}{7} + C \end{array}$

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Most popular questions from this chapter

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$\int \frac{\sqrt{4 - x^{2}}}{x^{2}} d x$

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

$\int \sin u d u$

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

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