Many integrals are most easily computed by means of a change of variables, commonly called a $u$-substitution.
Example
Let’s compute $\displaystyle\int\! 2x(x^2-1)^4\, dx$ by making the substitution \begin{eqnarray*} u&=&x^2-1\\ du&=&2x\, dx. \end{eqnarray*} Then \[ \int 2x(x^2-1)^4\, dx=\int (x^2-1)^4(2x\, dx)=\int u^4\, du=\frac{u^5}{5}+C=\frac{(x^2-1)^5}{5}+C.\] We may check this result by differentiating using the Chain Rule: \[\frac{d}{dx}\left(\frac{(x^2-1)^5}{5}+C\right)=\frac{5(x^2-1)^4}{5}(2x) =2x(x^2-1)^4.\qquad\qquad \surd\]
The substitution method amounts to applying the Chain Rule in reverse:
To compute $\displaystyle\int\! f(g(x))g'(x)\, dx$, we let \begin{eqnarray*} u&=&g(x)\\ du&=&g'(x)\, dx. \end{eqnarray*} Then \[\int f(g(x))g'(x)\, dx=\int f(u)\, du=F(u)=F(g(x))\] where $F$ is an antiderivative of $f$.
Example
To compute $\displaystyle\int\! \sin (2x)\cos (2x)\, dx$, let \begin{eqnarray*} u&=&\sin (2x)\\ du&=&2\cos (2x)\, dx. \end{eqnarray*} Then \[\int \sin (2x)\cos (2x)\, dx=\int\frac{1}{2}\sin (2x)[2\cos (2x)\, dx]=\int \frac{1}{2}u\, du=\frac{1}{4}u^2+C=\frac{1}{4}\sin^2 (2x)+C.\]
With practice, you will often be able to write down the result immediately.
Example
We can evaluate $\displaystyle\int\! \frac{dx}{(4x-3)^2}$ by letting \begin{eqnarray*} u&=&4x-3\\ du&=&4\, dx\quad\longrightarrow\quad dx=\frac{1}{4}\, du. \end{eqnarray*} Then \[\int \frac{dx}{(4x-3)^2}=\int \frac{\frac{1}{4}\, du}{u^2}=-\frac{1}{4u}+C=\frac{-1}{4(4x-3)}+C.\]
It is not always apparent until you try it whether or not a substitution will work.
Example
To compute $\displaystyle\int\! x\sqrt{x-3}\, dx$, we will try \begin{eqnarray*} u&=&x-3\quad\longrightarrow\quad x=u+3\\ du&=&dx. \end{eqnarray*} So \begin{eqnarray*} \int x\sqrt{x-3}\, dx&=&\int (u+3)\sqrt{u}\, du=\int \left(u^{3/2}+3u^{1/2}\right)\, du\\ &=&\frac{2}{5}u^{5/2}+2u^{3/2}+C=\frac{2}{5}(x-3)^{5/2}+2(x-3)^{3/2}+C. \end{eqnarray*}
We can also compute a definite integral using a substitution.
Example
Let’s evaluate $\displaystyle\int^2_0\! xe^{x^2}\, dx$. Let
\begin{eqnarray*}
u&=&x^2\\
du&=&2x\, dx.
\end{eqnarray*}
First, we will compute the indefinite integral:
\[\int xe^{x^2}\, dx=\int \left(\frac{1}{2}e^{x^2}\right)(2x\,
dx)=\int\frac{1}{2}e^u\, du=\frac{1}{2}e^u+C=\frac{1}{2}e^{x^2}+C.\]
Now we have two approaches for the definite integral:
Thus, we find that \[\int^2_0 xe^{x^2}\, dx=\frac{1}{2}(e^4-1).\]
Approach 2 works provided certain conditions on $f$ and $g$ are met: \[\int^b_a f(g(x))\, dx=\int^{g(b)}_{g(a)} f(u)\, du\] if
- $g’$ is continuous on $[a,b]$.
- $f$ is continuous on the set of values taken by $g$ on $[a,b]$.
Substitutions are useful or necessary for a huge range of integrals. You will find yourself either implicitly or explicitly using a substitution in virtually every integral you compute!
Key Concepts
The substitution method amounts to applying the Chain Rule in reverse:
To compute $\displaystyle\int\! f(g(x))g'(x)\, dx$, we let $u = g(x)$ $du = g'(x) dx$.
Then
$\displaystyle{\int f(g(x))g'(x)\, dx = \int f(u)\, du = F(u) = F(g(x))}$
where $F$ is an antiderivative of $f$.