Mathematics - Indefinite Integration Question with Solution | TestHub

The given integral is:

$$\int \frac{\sin 2x+2}{\sin 2x+\cos 2x+1}dx$$

We can rewrite the expression as:

$$\begin{array}{rl}& \int \frac{1+\sin x\cos x}{\sin x\cos x+{\cos }^{2}x}\\ & =\frac{1}{2}\int \frac{\left(1+\sin x\cos x+{\sin }^{2}x\right)+\left(1+\sin x\cos x-{\sin }^{2}x\right)}{\sin x\cos x+{\cos }^{2}x}dx\\ & =\frac{1}{2}\int \left(\frac{1+\sin x\cos x+{\sin }^{2}x}{\sin x\cos x+{\cos }^{2}x}+1\right)dx\\ & =\frac{1}{2}\int \left(\frac{{\sec }^{2}x}{1+\tan x}+\tan x+1\right)dx\\ & =\frac{1}{2}[\ln |1+\tan x|+\ln |\sec x|+x]+C\\ & =\frac{x}{2}+\frac{\ln |\sec x+\sec x\tan x|}{2}+C\end{array}$$

Given that $f(0)=0$:

$$\begin{array}{rl}& f(0)=0\\ & \therefore f(x)=\frac{x}{2}+\frac{\ln |\sec x+\sec x\tan x|}{2}\end{array}$$