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Evaluate the following integrals.

a) $\int^1_0 x e^x dx$

So integrating by parts we get

$u = x $ $vu = e^x dx$

$du = dx$ $ v = e^x$

$uv - \int vdu = x e^x - \int^1_0 e^x dx$

\(\displaystyle xe^x - e^x |^1_0 = 1\)

b) \(\displaystyle \int^1_0 x^2 e^x \, dx\)

Integrating by parts we get

\(\displaystyle u = x^2 \) \(\displaystyle dv = e^x dx\)

\(\displaystyle du = 2xdx\) \(\displaystyle v = e^x\)

\(\displaystyle uv - \int vdu = x^2 e^x - \int^1_0 e^x 2x = e^1 - 2 \)

a) $\int^1_0 x e^x dx$

So integrating by parts we get

$u = x $ $vu = e^x dx$

$du = dx$ $ v = e^x$

$uv - \int vdu = x e^x - \int^1_0 e^x dx$

\(\displaystyle xe^x - e^x |^1_0 = 1\)

b) \(\displaystyle \int^1_0 x^2 e^x \, dx\)

Integrating by parts we get

\(\displaystyle u = x^2 \) \(\displaystyle dv = e^x dx\)

\(\displaystyle du = 2xdx\) \(\displaystyle v = e^x\)

\(\displaystyle uv - \int vdu = x^2 e^x - \int^1_0 e^x 2x = e^1 - 2 \)

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