/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 18

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

Short Answer

Expert verified
Question: Evaluate the integral of the given function: $$\int \frac{\sin ^{3} x}{\cos ^{5} x} d x$$ Solution: Using the substitution method and simplifying the integral, we find that the integral evaluates to: $$\frac{1}{4} \cos^4 x + \frac{3}{2} \cos^2 x + \frac{1}{2} \cos^2 x + C$$

Step by step solution

01

Choose a suitable substitution

We will let \(u = \cos x\). Therefore, \(du = -\sin x \, dx\).
02

Express the sine function in terms of cosine function

As we know \(\sin^2 x + \cos^2 x = 1\), we find that \(\sin^2 x = 1 - \cos^2 x\). We will need this relation to replace the sine function in the given integral.
03

Replace the variables in the integral

We have \(\int \frac{\sin ^{3} x}{\cos ^{5} x} d x\). We will replace the variables using the substitution, \(u = \cos x\). The integral will now be: $$\int \frac{(1 - u^2)^{\frac{3}{2}}}{u^5} (-du)$$
04

Simplify the integral

Now, we will simplify the integral, taking the negative sign outside: $$-\int \frac{(1 - u^2)^{\frac{3}{2}}}{u^5} du$$
05

Expand the integrand and rewrite the integral

Expanding \((1 - u^2)^{\frac{3}{2}}\) yields, $$\left(1 - 3u^2 - (-3 u^2) + u^6 \right)$$ We now rewrite the integral, $$-\int \frac{1 - 3u^2 + u^6}{u^5} du$$
06

Separate the fractions and simplify

We will rewrite the integrand as the sum of three separate fractions: $$-\left(\int \frac{1}{u^5} du - 3\int \frac{u^2}{u^5} du + \int \frac{u^6}{u^5} du\right)$$ Now we simplify the integrals: $$-\left(\int u^{-5} du - 3\int u^{-3} du + \int u^1 du\right)$$
07

Integrate each expression

Now, we will integrate each expression individually and then combine the results: $$-\left(\frac{u^{-4}}{-4} - \frac{3u^{-2}}{-2} + \frac{u^2}{2} + C\right)$$
08

Substitute back the original variable and simplify

Now replace the variable \(u\) with the original variable, \(x\): $$-\left(\frac{\cos^{-4} x}{-4} - \frac{3\cos^{-2} x}{-2} + \frac{\cos^2 x}{2} + C\right)$$ Now, we simplify the expression: $$\frac{1}{4} \cos^4 x + \frac{3}{2} \cos^2 x + \frac{1}{2} \cos^2 x + C$$ So, the evaluated integral of the given function is: $$\int \frac{\sin ^{3} x}{\cos ^{5} x} d x = \frac{1}{4} \cos^4 x + \frac{3}{2} \cos^2 x + \frac{1}{2} \cos^2 x + C$$

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Most popular questions from this chapter

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