Integrate $\int\frac{e^{\arcsin x} + x + 1}{\sqrt{1-x^2}}dx$

Question

does anybody have an advice how I can integrate this $$\int\frac{e^{\arcsin x} + x + 1}{\sqrt{1-x^2}}dx$$ I tried substitution $\arcsin x=t$, but was not able to finish it.

Answer 1

I got this. When you substitute $arcsinx=t$ you got really good output. $$\int\frac{e^{arcsinx}}{\sqrt{1-x^2}}dx$$ if $arcsinx=t$ => $dt=\frac{dx}{\sqrt{1-x^2}}$ and the integral transforms into $$\int e^tdt=e^{arcsinx} + C_1$$ The whole answer is $$e^{arcsinx} -\sqrt{1-x^2} + arcsinx +C$$ Thats it.