Solve ∫ (from 0 to frac { pi) of {2}}arcsinx wrt x+(1-frac{sqrt{2}}{2})pi/2-int_{pi/2}^1arcsinxdx

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \arcsin x d x + ( 1 - \frac { \sqrt { 2 } } { 2 } ) \frac { \pi } { 2 } - \int _ { \frac { \pi } { 2 } } ^ { 1 } \arcsin x d x$$

Factor

$$\frac{4\int _{0}^{\frac{\pi }{2}}\arcsin(x)\mathrm{d}x+4\int _{1}^{\frac{\pi }{2}}\arcsin(x)\mathrm{d}x+\pi \sqrt{2}\left(\sqrt{2}-1\right)}{4}$$

Solution Steps

Factor out \frac{1}{4}.

$$\frac{4\int _{0}^{\frac{\pi }{2}}\arcsin(x)\mathrm{d}x+\sqrt{2}\left(\sqrt{2}-1\right)\pi -4\int _{\frac{\pi }{2}}^{1}\arcsin(x)\mathrm{d}x}{4}$$

Evaluate

$$\frac{4\int _{0}^{\frac{\pi }{2}}\arcsin(x)\mathrm{d}x+4\int _{1}^{\frac{\pi }{2}}\arcsin(x)\mathrm{d}x-\pi \sqrt{2}+2\pi }{4}$$