Indefinite Integral $\cos(x^2)\sin(e^{x^2})$

Question

$$\int \sin \left(e^{x^2}\right) \cos\left(x^2\right) \, dx$$

I have the substitution $u = x^2$ and $du = 2x \, dx$

I have no idea what to do after this. Any help would be appreciated.

Answer 1

By integration of parts assuming that $v' = 1$ and $u= \sin(e^{x^2})\cos(x^2)$ $$\int \sin(e^{x^2})\cos(x^2)dx = x\sin(e^{x^2})\cos(x^2) \space + \space\int 4x^3 \sin(x^2) \cos(e^{x^2}) dx$$ Using the substitution $u = x^2$ the RHS becomes

$$x\sin(e^{x^2})\cos(x^2) + \int2u\sin(u)\cos(e^u)du$$

This is what the integrand would look like after applying the substituion, if this was your question.

EDIT: for the sake of placing it all under one integrand i have made this edit.

For us to place it all under one integrand the variable must be the same so we use $x = \pm \root\of{u}$ and differentiate the terms outside the intgeral with resepect to $u$

$$x\sin(e^{x^2})\cos(x^2) + \int2u\sin(u)\cos(e^u)du$$ $$= \pm\int \frac{d}{du}(\root \of{u}\sin(e^u)\cos(u))du +\int2u\sin(u)\cos(e^u)du$$

$$= \int \pm\frac{1}{2\root \of{u}}\sin(e^u)\cos(u) \mp\root \of{u}\sin(e^u)\sin(u) \pm \root \of{u}e^u\cos(e^u)\cos(u) +2u\sin(u)\cos(e^u)du$$