Question

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

Answer 1

`=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})|`