Solve ∫ frac{sqrt{2}sinx}{sin(x-pi/4) wrt x=

$$\int \frac { \sqrt { 2 } \sin x } { \sin ( x - \frac { \pi } { 4 } ) } d x =$$

Evaluate

$$\frac{\sqrt{2}x\sin(x)}{\sin(\frac{4x-\pi }{4})}-x\cot(\frac{4x-\pi }{4})+\ln(|\sin(\frac{4x-\pi }{4})|)+С$$

Solution Steps

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\sqrt{2}\int \frac{\sin(x)}{\sin(\frac{1}{4}\left(4x-\pi \right))}\mathrm{d}x$$

Simplify.

$$\sqrt{2}\left(\frac{\sqrt{2}\ln(|\sin(x-\frac{\pi }{4})|)}{2}+\frac{x\sin(x)-\frac{x\sqrt{2}\cos(x-\frac{\pi }{4})}{2}}{\sin(x-\frac{\pi }{4})}\right)$$

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

$$\left(\frac{\sqrt{2}\ln(|\sin(x-\frac{\pi }{4})|)}{2}+\frac{x\sin(x)-\frac{x\sqrt{2}\cos(x-\frac{\pi }{4})}{2}}{\sin(x-\frac{\pi }{4})}\right)\sqrt{2}+С$$

Differentiate w.r.t. x

$$\frac{\sqrt{2}\sin(x)\sin(\frac{4x-\pi }{4})-\sqrt{2}\sin(\frac{\pi }{4})x+x}{\left(\sin(\frac{4x-\pi }{4})\right)^{2}}$$