∫ 1/1+tanxdu eching

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

Baholash

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

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

Yechim qadamlari

\int a\mathrm{d}u=au umumiy integrallar qoidasi jadvalidan foydalanib, \frac{1}{1+\frac{\sin(x)}{\cos(x)}} integralini toping.

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

Qisqartirish.

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

Агар F\left(u\right)f\left(u\right) ning dastlabki holati boʻlsa, u holatda f\left(u\right) ning barcha dastlabki holatlari toʻplami F\left(u\right)+C tarafidan belgilanadi. Shu sababli natijaga C\in \mathrm{R} integrallash konstantasini qoʻshing.

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

x ga nisbatan hosilani topish

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