Problem 42

Question

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

Step-by-Step Solution

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Answer

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

1Step 1: 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.

2Step 2: 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 $-\int u^{-3} du$.

3Step 3: 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: \[-\frac{1}{2} \times u^{-2}\].

4Step 4: Back Substitute

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

5Step 5: 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 $-1/(2\cos^2(x)) + C$.

Problem 42

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