Solve 3-x/2sqrt{1-2x-x^2}+2arcsin(1+x/sqrt{2})

$$\frac{ 3-x }{ 2 } \sqrt{ 1-2x- { x }^{ 2 } } +2 \arcsin ( \frac{ 1+x }{ \sqrt{ 2 } } )$$

Factor

$$\frac{4\arcsin(\frac{\sqrt{2}\left(x+1\right)}{2})+\left(3-x\right)\sqrt{1-2x-x^{2}}}{2}$$

Solution Steps

Express \frac{3-x}{2}\sqrt{1-2x-x^{2}} as a single fraction.

$$factor(\frac{\left(3-x\right)\sqrt{1-2x-x^{2}}}{2}+2\arcsin(\frac{1+x}{\sqrt{2}}))$$

Rationalize the denominator of \frac{1+x}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$factor(\frac{\left(3-x\right)\sqrt{1-2x-x^{2}}}{2}+2\arcsin(\frac{\left(1+x\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}))$$

The square of \sqrt{2} is 2.

$$factor(\frac{\left(3-x\right)\sqrt{1-2x-x^{2}}}{2}+2\arcsin(\frac{\left(1+x\right)\sqrt{2}}{2}))$$

Use the distributive property to multiply 3-x by \sqrt{1-2x-x^{2}}.

$$factor(\frac{3\sqrt{1-2x-x^{2}}-x\sqrt{1-2x-x^{2}}}{2}+2\arcsin(\frac{\left(1+x\right)\sqrt{2}}{2}))$$

Use the distributive property to multiply 1+x by \sqrt{2}.

$$factor(\frac{3\sqrt{1-2x-x^{2}}-x\sqrt{1-2x-x^{2}}}{2}+2\arcsin(\frac{\sqrt{2}+x\sqrt{2}}{2}))$$

Factor out \frac{1}{2}.

$$\frac{3\sqrt{1-2x-x^{2}}-x\sqrt{1-2x-x^{2}}+4\arcsin(\frac{\sqrt{2}\left(1+x\right)}{2})}{2}$$

Evaluate

$$\frac{4\arcsin(\frac{\sqrt{2}\left(x+1\right)}{2})+\left(3-x\right)\sqrt{1-2x-x^{2}}}{2}$$