Evaluate $\int(2x + 9)e^{x^2+9x}dx$ using u-substitution

Question

5.3 Note: Webassign has a typo here, i messaged the guy to get him to fix it. Problem #4 should read:

Evaluate $\int(2x + 9)e^{x^2+9x}dx$ using u-substitution

Can somebody verify this for me?

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Solution: Let $u=x^2+9x$. Then $\frac{du}{dx}=2x+9$ and so $\frac{du}{2x+9}=dx$.

Thus we have:

$\int(2x + 9)e^{x^2+9x}dx$

$= \int(2x + 9)e^{u}\frac{du}{2x+9}$

$= \int e^udu$

$=e^u +C$

$=e^{x^2+9x}+C$

Answer 1

Looks good to me! That's the correct answer.

Answer 2

This one is custom made, since we have $f'(x)e^ {f(x)}$. Just substitute $u=x^2+9x$, as you have done. Since $e^x$ is its own derivative, by the chain rule everything works out.