Question

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

Short answer

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

Step-by-step solution

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.

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 \(-\int u^{-3} du\).

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: \[-\frac{1}{2} \times u^{-2}\].

Back Substitute

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

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 \(-1/(2\cos^2(x)) + C\).