Question

Q. Intigrate
ʃ(sinx+cosx/tanx)dx

Answer 1

Answer on Question #53147 – Math – Integral Calculus

Question

Integrate $\int \left(\sin x + \frac{\cos x}{\tan x}\right) dx$.

Solution

$$\begin{array}{l} \int \left(\sin x + \frac{\cos x}{\tan x}\right) dx = \int \left(\sin x + \frac{(\cos x)^2}{\sin x}\right) dx = \int \frac{dx}{\sin x} = \int \frac{\sin x}{(\sin x)^2} dx = \\ = -\int \frac{d(\cos x)}{1 - (\cos x)^2} = \left\{\cos x = u\right\} = -\int \frac{du}{1 - u^2} = \int \frac{du}{u^2 - 1} = \frac{1}{2} \ln \left| \frac{u - 1}{u + 1} \right| + c = \frac{1}{2} \ln \left| \frac{\cos x - 1}{\cos x + 1} \right| + c, \end{array}$$

where $c$ is an arbitrary real constant.

Answer. $\frac{1}{2} \ln \left| \frac{\cos x - 1}{\cos x + 1} \right| + c$.

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