Integration by Parts

Let $f(x)$ and $g(x)$ be differentiable functions. Then the product rule
$$(f(x)g(x))’=f’(x)g(x)+f(x)g’(x)$$
leads to the integration
\begin{equation}
\label{eq:intpart}
\int f(x)g’(x)dx=f(x)g(x)-\int f’(x)g(x)dx.
\end{equation}
The formula \eqref{eq:intpart} is called integration by parts. If we set $u=f(x)$ and $v=g(x)$, then \eqref{eq:intpart} can be also written as
\begin{equation}
\label{eq:intpart2}
\int udv=uv-\int vdu.
\end{equation}

Example. Evaluate $\int x\cos xdx$.

Solution. Let $u=x$ and $dv=\cos xdx$. Then $du=dx$ and $v=\sin x$. So,
\begin{align*}
\int x\cos xdx&=x\sin x-\int\sin xdx\\
&=x\sin x+\cos x+C,
\end{align*}
where $C$ is a constant.

Example. Evaluate $\int\ln xdx$.

Solution. Let $u=\ln x$ and $dv=dx$. Then $du=\frac{1}{x}dx$ and $v=x$. So,
\begin{align*}
\int\ln xdx&=x\ln x-\int x\cdot\frac{1}{x}dx\\
&=x\ln x-x+C,
\end{align*}
where $C$ is a constant.

Often it is required to apply integration by parts more than once to evaluate a given integral. In that case, it is convenient to use a table as shown in the following example.

Example. Evaluate $\int x^2e^xdx$

Solution. In the following table, the first column represents $x^2$ and its derivatives, and the second column represents $e^x$ and its integrals.
$$\begin{array}{ccc}
x^2 & & e^x\\
&\stackrel{+}{\searrow}&\\
2x & & e^x\\
&\stackrel{-}{\searrow}&\\
2 & & e^x\\
&\stackrel{+}{\searrow}&\\
0 & & e^x.
\end{array}$$
This table shows the repeated application of integration by parts. Following the table, the final answer is given by
$$\int x^2e^xdx=x^2e^x-2xe^x+2e^x+C,$$
where $C$ is a constant.

Example. Evaluate $\int x^3\sin xdx$.

Solution. In the following table, the first column represents $x^3$ and its derivatives, and the second column represents $\sin x$ and its integrals.
$$\begin{array}{ccc}
x^3 & & \sin x\\
&\stackrel{+}{\searrow}&\\
3x^2 & & -\cos x\\
&\stackrel{-}{\searrow}&\\
6x & & -\sin x\\
&\stackrel{+}{\searrow}&\\
6 & & \cos x\\
&\stackrel{-}{\searrow}&\\
0 & & \sin x.
\end{array}$$
Following the table, the final answer is given by
$$\int x^3\sin xdx=-x^3\cos x+3x^2\sin x+6x\cos x-6\sin x+C,$$
where $C$ is a constant.

Example. Evaluate $\int e^x\cos xdx$.

Solution. In the following table, the first column represents $e^x$ and its derivatives, and the second column represents $\cos x$ and its integrals.
$$\begin{array}{ccc}
e^x & & \cos x\\
&\stackrel{+}{\searrow}&\\
e^x & & \sin x\\
&\stackrel{-}{\searrow}&\\
e^x & & -\cos x.
\end{array}$$
Now, this is different from the previous two examples. While the first column repeats the same function $e^x$, the functions second column changes from $\cos x$ to $\sin x$ and to $\cos x$ again up to sign. In this case, we stop there and write the answer as we have done in the previous two examples and add to it $\int e^x(-\cos x)dx$. (Notice that the integrand is the product of functions in the last row.) That is,
$$\int e^x\cos xdx=e^x\sin x-e^x\cos x-\int e^x\cos xdx.$$
For now we do not worry about the constant of integration. Solving this for $\int e^x\cos xdx$, we obtain the final answer
$$\int e^x\cos xdx=\frac{1}{2}e^x\sin x-\frac{1}{2}e^x\cos x+C,$$
where $C$ is a constant.

Example. Evaluate $\int e^x\sin xdx$.

Solution. In the following table, the first column represents $e^x$ and its derivatives, and the second column represents $\sin x$ and its integrals.
$$\begin{array}{ccc}
e^x & & \sin x\\
&\stackrel{+}{\searrow}&\\
e^x & & -\cos x\\
&\stackrel{-}{\searrow}&\\
e^x & & -\sin x.
\end{array}$$
This is similar to the above example. The first columns repeats the same function $e^x$, and the functions in the second column changes from $\sin x$ to $\cos x$ and to $\sin x$ again up to sign. So we stop there and write
$$\int e^x\sin xdx=-e^x\cos x+e^x\sin x-\int e^x\sin xdx.$$
Solving this for $\int e^x\sin xdx$, we obtain
$$\int e^x\sin xdx=-\frac{1}{2}e^x\cos x+\frac{1}{2}e^x\sin x+C,$$
where $C$ is a constant.

Example. Evaluate $\int e^{5x}\cos 8xdx$.

Solution. In the following table, the first column represents $e^{5x}$ and its derivatives, and the second column represents $\cos 8x$ and its integrals.
$$\begin{array}{ccc}
e^{5x} & & \cos 8x\\
&\stackrel{+}{\searrow}&\\
5e^{5x} & & \frac{1}{8}\sin 8x\\
&\stackrel{-}{\searrow}&\\
25e^{5x} & & -\frac{1}{64}\cos 8x.
\end{array}$$
The first columns repeats the same function $e^{5x}$ up to constant multiple, and the functions in the second column changes from $\cos 8x$ to $\sin 8x$ and to $\cos 8x$ again to constant multiple. This case also we do the same.
$$\int e^{5x}\cos 8xdx=\frac{1}{8}e^{5x}\sin 8x+\frac{5}{64}e^{5x}\cos 8x-\frac{25}{64}\int e^{5x}\cos 8xdx.$$
Solving this for $\int e^{5x}\cos 8xdx$, we obtain
$$\int e^{5x}\cos 8xdx=\frac{8}{89}e^{5x}\sin 8x+\frac{5}{89}e^{5x}\cos 8x+C,$$
where $C$ is a constant.

The evaluation of a definite integral by parts can be done as
\begin{equation}
\label{eq:intpart3}
\int_a^b f(x)g’(x)dx=[f(x)g(x)]_a^b-\int_a^b f’(x)g(x)dx.
\end{equation}

Example. Find the area of the region bounded by $y=xe^{-x}$ and the x-axis from $x=0$ to $x=4$.

The graph of y=xexp(-x), x=0..4

The graph of y=xexp(-x), x=0..4

Solution. Let $u=x$ and $dv=e^{-x}dx$. Then $du=dx$ and $v=-e^{-x}$. Hence,
\begin{align*}
A&=\int_0^4 xe^{-x}dx\\
&=[-xe^{-x}]0^4+\int_0^4 e^{-x}dx\\
&=-4e^{-4}+[-e^{-x}]_0^4\\
&=1-5e^{-4}.
\end{align*}

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