What is the antiderivative of (cos^2x) / sinx?

1 Answer
May 22, 2016

=cos (x)+\ln |tan (frac{x}{2})|

Explanation:

\int \frac{\cos ^2(x)}{\sin \(x)}dx

We know,
\cos ^2(x)=1-\sin ^2(x)

=\int \frac{1-\sin ^2(x)}{\sin (x)}dx

Applying sum rule,
\int f(x)\pm g(x)dx=\int f(x)dx\pm \int g(x)dx

=\int \frac{1}{\sin (x)}dx-\int \frac{\sin ^2(x)}{\sin (x)}dx

Also,we know,
\int \frac{1}{\sin (x)}dx=\ln |\tan \(\frac{x}{2})\|

\int \frac{\sin ^2(x)}{\sin \(x)}dx=-\cos (x)

now,
=\ln |\tan (\frac{x}{2})|-\(-\cos (x))

simplifying it,
=cos (x)+\ln |tan (frac{x}{2})|