Note: The following steps ignore the bounds of integration. Be sure to apply them at the end.
Let $f = x^{2}$ and $dg = e^xdx$.
Then $df = 2x dx$ and $g = e^{x}$.

$x^2e^x-\int2xe^xdx$

For the integral:
Let $f = 2x$ and $dg = e^{x}dx$.
Then $df = 2dx$ and $g = e^{x}$.

$x^2e^x-\Big(2xe^x-\int2e^xdx\Big)$

You can now integrate the integral that is left without using integration by parts.

Use the same process outlined in part (a), but use $e^{3x}$ instead of $e^{x}$.

Let $f = x$ and $dg = \sin\left(x\right)$.

Let $f = \sin\left(x\right)$ and $dg = \sin\left(x\right)dx$.

$-\sin(x)\cos(x)+\int\cos^2(x)dx$

$=-\sin(x)\cos(x)+\int(1-\sin^2(x))dx$

$\int\sin^2(x)dx=-\sin(x)\cos(x)+x-\int\sin^2(x)dx$

Now solve this equation for the original integral.