Question

Evaluate the integral. $$ \int \frac{\cos ^{5} x}{\sin ^{3} x} d x $$

   Evaluate the integral.
$$
\int \frac{\cos ^{5} x}{\sin ^{3} x} d x
$$

Calculus for AP

Calculus for AP

Jon Rogawski & Colin… 3rd Edition

Chapter 7, Problem 31

Instant Answer

Solved by Expert Suman Saurav Thakur

11/09/2020

Step 1

Step 1: Rewrite the integral in terms of cosine and sine functions: $$ \int \frac{\cos ^{5} x}{\sin ^{3} x} d x = \int \frac{(\cos^2 x)^2 \cos x}{\sin^3 x} dx = \int \frac{(1-\sin^2 x)^2 \cos x}{\sin^3 x} dx $$ Show more…

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Evaluate the integral. $$ \int \frac{\cos ^{5} x}{\sin ^{3} x} d x $$

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Key Concepts

Substitution Method

This is an integration technique where a new variable is introduced to simplify the integrand. In problems where a function and its derivative are present, substituting a part of the expression (such as setting u equal to a trigonometric function) can convert the integral into a more manageable form, making the integration process straightforward.

Trigonometric Identities

These are equations involving trigonometric functions that are true for all values in their domains. They are often used to rewrite powers or products of sine and cosine functions into a simpler form. In integration, such identities help in reducing the powers or transforming the integrand into a form that is easier to integrate.

Integration of Rational Functions

After applying substitution and rewriting trigonometric expressions using identities, integrals can often be transformed into rational functions. Integrating rational functions involves techniques like division, partial fraction decomposition, or straightforward term-by-term integration, which are central strategies in calculus.

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