Calculate the integral $\int_{0}^{\frac{\pi}{2}} \cos(x) \left( \int_{1}^{\sin(x)} e^{t^2} dt \right) dx$

Question

Calculate the following integral:$\int_{0}^{\frac{\pi}{2}} \cos(x) \left( \int_{1}^{\sin(x)} e^{t^2} dt \right) dx$.
I tried integrating by parts but ended up getting Gauss error function (which hasn't been contemplated by my course yet).Is there a way to calculate the integral without going through the gauss function?Any help would be kindly appreciated.

Answer 1

Integrate by parts

\begin{align} &\int_{0}^{\frac{\pi}{2}} \cos x \left( \int_{1}^{\sin2x} e^{t^2} dt \right) dx\\ =& \int_{0}^{\frac{\pi}{2}} \left( \int_{1}^{\sin x } e^{t^2} dt \right) d(\sin x) =-\int_0^{\frac\pi2}e^{\sin^2 x}\sin x\cos x\ \overset{y=2x}{dx}\\ =&\ \frac14 e^{\frac12}\int_0^\pi e^{-\frac12\cos y}\ d(\cos y) =\frac12(1-e) \end{align}