Problems on Techniques of Integration

The exponential function is a nice function since its derivative and antiderivative are the same. So for the integration by parts, this function has the same behavior whether we differentiate it or take its antiderivative. Therefore the focus should be on the other function involved in the integration. In this case, we must differentiate $f(x) = x$ because its derivative gives the constant 1. We may also use the antiderivative of the function $xe^x$ and evaluate it at the points $0$ and $1$.

Set

\begin{displaymath}\left\{\begin{array}{lll} u &=&x\\ dv &=& e^{x}dx\;. \end{array}\right.\end{displaymath}

Then

\begin{displaymath}\left\{\begin{array}{lll} du &=& dx\\ v &=& e^{x}\;. \end{array}\right.\end{displaymath}

Since

\begin{displaymath}\int_a^b u dv = [u v]_a^b - \int_a^b v du\;,\end{displaymath}

we get

\begin{displaymath}\int_0^1 x e^x dx = [x e^x]_0^1 - \int_0^1 e^x dx\end{displaymath}

which implies

\begin{displaymath}\int_0^1 x e^x dx = [x e^x]_0^1 - [e^x]_0^1 = (e - 0) - (e-1) = 1\;.\end{displaymath}

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