Questions: Consider the following integral [ int fracsin (x)cos ^4(x) d x ] - Which of the following methods of integration is the best for this antiderivative problem? Elementary Antiderivative Integration by Substitution Integration by Parts Trigonometric Substitution Integration by Partial Fractions

$Consider the following integral [ int fracsin (x)cos ^4(x) d x ] - Which of the following methods of integration is the best for this antiderivative problem? Elementary Antiderivative Integration by Substitution Integration by Parts Trigonometric Substitution Integration by Partial Fractions$

Transcript text: b. Consider the following integral \[ \int \frac{\sin (x)}{\cos ^{4}(x)} d x \] - Which of the following methods of integration is the best for this antiderivative probl Elementary Antiderivative Integration by Substitution Integration by Parts Trigonometric Substitution Integration by Partial Fractions - Enter in the additional information.

Solution

Solution Steps

Solution Approach

The best method for this integral is Integration by Substitution. We can use the substitution $ u = \cos(x) $, which simplifies the integral.

Step 1: Identify the Integral and Choose the Method

We are given the integral: \[ \int \frac{\sin(x)}{\cos^4(x)} \, dx \] The best method for this integral is Integration by Substitution.

Step 2: Perform the Substitution

Let $ u = \cos(x) $. Then, $ du = -\sin(x) \, dx $ or $ dx = -\frac{du}{\sin(x)} $.

Step 3: Rewrite the Integral

Substitute $ u $ and $ du $ into the integral: \[ \int \frac{\sin(x)}{\cos^4(x)} \, dx = \int \frac{\sin(x)}{u^4} \left(-\frac{du}{\sin(x)}\right) = -\int \frac{1}{u^4} \, du \]

Step 4: Integrate with Respect to $ u $

Integrate $ -\int \frac{1}{u^4} \, du $: \[ -\int u^{-4} \, du = -\left( \frac{u^{-3}}{-3} \right) = \frac{u^{-3}}{3} = \frac{1}{3u^3} \]

Step 5: Substitute Back $ u = \cos(x) $

Replace $ u $ with $ \cos(x) $: \[ \frac{1}{3u^3} = \frac{1}{3\cos^3(x)} \]

Step 6: Simplify the Result

The simplified result is: \[ -\frac{\tan^3(x)}{3} \]