Calculus I Evaluation of Definite Integral

Question

There is a problem I am having trouble understanding.

We are asked to evaluate the definite integral:

$$\int_0^2\sin(e^x+x^2)(e^x+2x)\,dx$$

If anyone would be so kind as to walk me through it, I would be extremely grateful.

Answer 1

Hint

Make the substitution $$ u=e^x+x^2 $$ and then use the fact that $$ \frac{d}{dx}(-\cos x)=\sin x $$

Answer 2

With the substitution $u(x) = e^x + x^2$, you get $$du = (e^x + 2x) dx $$ and so your integral becomes $$\int_{u(0)}^{u(2)} \sin u\ du = [-\cos u]_{1}^{e^2 + 4} $$