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Chapter 5: Q 40. (page 452)

Solve the following integral.
$鈭� \sin^{2} x \cos^{3} x d x$

Short Answer

Expert verified

Answer is $\frac{\sin^{3} x}{3} - \frac{\sin^{5} x}{5} + C$

Step by step solution

01

Step 1. Given information

An integral is $鈭� \sin^{2} x \cos^{3} x d x$

02

Step 2. Explanation

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

Let $u = \sin x$

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

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Most popular questions from this chapter

Consider the integral $鈭� x {(x^{2} 鈭� 1)}^{2} d x$ .

(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.

Solve the integral: $鈭� x \ln (x) d x$

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.

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

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

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

Solve $鈭� \frac{x + 2}{{(x^{2} + 4 x)}^{\frac{3}{2}}} d x$ the following two ways:

(a) with the substitution $u = x^{2} + 4 x;$

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.

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