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Chapter 5: Q 15. (page 451)

Describe strategy for solving the type of integral given.
$鈭� \sin^{m} x \cos^{n} x d x$ , both $m, n$ even

Short Answer

Expert verified

Write $\sin^{m} x = \sin^{2} x \sin^{m - 2} x$ or $\cos^{n} x = \cos^{2} x \cos^{n - 2} x$

Step by step solution

01

Step 1. Given information

Integral is $鈭� s i n^{m} x c o s^{n} x d x$ , both $m, n$ even

02

Step 2. Explanation

Consider the integral $鈭� \sin^{m} x \cos^{n} x d x$ , both $m, n$ even

Use the formula $\sin^{2} x = 1 - \cos^{2} x$ or $\cos^{2} x = 1 - \sin^{2} x$

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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 given definite integral.

${鈭�}_{1 / 4}^{1 / 2} 鈥� \frac{1}{x^{2} \sqrt{1 鈭� x^{2}}} d x$

Find three integrals in Exercises 21鈥�70 that we can anti-differentiate immediately after algebraic simplification.

Solve the integral: $鈭� x \ln (x^{2}) d x$

Solve given definite integral.

${鈭�}_{1}^{2} 鈥� \frac{1}{x \sqrt{9 鈭� x^{2}}} d x$

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