Problem 20 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \frac{\cos x}{\sqrt{\sin ^{2} x+1}} d x $$

Step-by-Step Solution

Answer

The result of the integral \(\int \frac{\cos x}{\sqrt{\sin^2 x+1}} dx\) is \(\ln |\sin(x) + \sqrt{\sin^2(x)+1}| + C\).

Step 1: Substitution

Set \(t = \sin(x)\). The differential of \(t\) is \(dt = \cos(x) \, dx\). Substitute these into the integral, we get\[\int \frac{dt}{\sqrt{t^2+1}}\]

Step 2: Use integration table

Use the integration table or know that \(\int \frac{dt}{\sqrt{t^2+1}} = \ln |t + \sqrt{t^2+1}|\). Therefore, our integral becomes \[\ln |t + \sqrt{t^2+1}| + C\]

Step 3: Substitute back original variable

Finally, substitute \(t = \sin(x)\) back into the solution \[\Rightarrow \ln |\sin(x) + \sqrt{\sin^2(x)+1}|+ C\]