/*! 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 42 Find the indefinite integral. ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 42

Find the indefinite integral. $$ \int \frac{\sin x}{\cos ^{3} x} d x $$

Short Answer

Expert verified
The indefinite integral is \(-1/(2\cos^2(x)) + C\).

Step by step solution

01

Choose the Substitution

Choose \(u = \cos(x)\) for the substitution. Differentiate \(u\) with respect to \(x\) to find \(du\), the differential of \(u\). This gives us, \(du =- \sin(x)dx \). Notice that this substitution allows us to replace both \(\sin(x)dx\) and \(\cos^3(x)\) in the original integral.
02

Substitute

Replace \(\sin(x)dx\) and \(\cos^3(x)\) in the integral with \( -du \) and \( u^3 \) respectively. The integral thus becomes \(-\int 1/u^3 du \), which simplifies to \(-\int u^{-3} du\).
03

Evaluate the Integral

Use the rule of integration saying the indefinite integral of \(u^n\) with respect to \(u\) is given by \((u^{n+1})/(n+1)\) for \(n \neq -1\) where \(n\) is any real number. This gives us: \[-\frac{1}{2} \times u^{-2}\].
04

Back Substitute

Finally, replace \(u\) with original function which was \(\cos(x)\), giving us \(-1/(2\cos^2(x))\).
05

Add Integration Constant

The integral of a function is not a single function, but a family of functions differing by a constant (\(C\)). Thus, the most general integral is \(-1/(2\cos^2(x)) + C\).

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Most popular questions from this chapter

Find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x^{2}} \sin \theta^{2} d \theta $$

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