Solve ∫ 1/1+tanxdu

$$\int \frac { 1 } { 1 + \tan x } d u$$

Evaluate

$$\frac{u\cos(x)}{\sin(x)+\cos(x)}+С$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{3\pi }{4}$$

Solution Steps

Find the integral of \frac{1}{1+\frac{\sin(x)}{\cos(x)}} using the table of common integrals rule \int a\mathrm{d}u=au.

$$\frac{u}{1+\frac{\sin(x)}{\cos(x)}}$$

Simplify.

$$\frac{\cos(x)u}{\cos(x)+\sin(x)}$$

If F\left(u\right) is an antiderivative of f\left(u\right), then the set of all antiderivatives of f\left(u\right) is given by F\left(u\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\begin{matrix}\frac{\cos(x)u}{\cos(x)+\sin(x)}+С_{3},&\end{matrix}$$

Differentiate w.r.t. x

$$-\frac{u}{\left(\sin(x)+\cos(x)\right)^{2}}$$