section 4.5

Substitution

By the way, I may call this variously “u-substitution” (which most people do) or “basic substitution.” When I call it basic, I don’t mean to be rude, it’s just that I’m teaching calc 2 at the same time, and in that class I’ve been saying “this is a basic calc 1 substitution” a lot.

Recall: The Chain Rule #

Recall: If $h(x) = f(g(x))$, then $$ h’(x) = f’(g(x)) g’(x) $$

Since integration is the inverse operation of differentiation, let’s integrate both sides with respect to x:

$\begin{align*} \int h’(x) \, dx &= \int f’(g(x)) g’(x)\, dx \ &= f(g(x)) + C \end{align*}$

Here I’m underlining that part involving the composed inner function to really drive home the observation:

$$ \int f’(\underline{g(x)}) \underline{g’(x)}\, dx = f(\underline{g(x)}) + C $$

Integration by the Substitution Rule #

Example #

Evaluate the integral $$ \int 2x \sqrt{1+x^2} \, dx $$

Solution We identify that there is an outer function and an inner function in a composition. We’re going to substitute $u = 1+x^2$. Then, using differentials, we’ll write $$ du = 2x \, dx $$

Then our original integral will become: $\begin{align*} \int 2x \sqrt{1+x^2} \, dx &= \int \sqrt{1+x^2} 2x \, dx \ &= \int \sqrt{u} \, du \ &= \int u^{1/2} \, du \ &= \frac{u^{3/2}}{3/2} + C \ &= \frac{2 u^{3/2}}{3} +C \end{align*}$

but our original integral was defined in terms of $x$ not $u$, so we substitute back to find $$ \int 2x \sqrt{1+x^2} \, dx = \frac{2 (1+x^2)^{3/2}}{3} +C $$

Did we get it right? You can always check your answer by differentiating! Here I’m asking WolframAlpha to perform the calculation to check my work.

Example #

Evaluate $\displaystyle \int x \sin (x^2) \,dx$

Example #

Evaluate $\displaystyle \int \sqrt{1 - 2x} \,dx$

Example #

Evaluate $\displaystyle \int \frac{x}{\sqrt{x^2 - 1}} \, dx $

Definite Integrals and substitution #

If $g’$ is continuous on $[a, b]$ and $f$ is continuous on the range of $g$ and $u=g(x)$, then: $$ \int_a^b f(g(x))g’(x)\, dx = \int_{g(a)}^{g(b)} f(u) \, du $$

that is, when we change the variable via a substitution, we also change the bounds. Let’s see some examples.

Example #

Evaluate $\displaystyle \int_1^2 \dfrac{dx}{(1-3x)^2}$.

Example #

Evaluate $\displaystyle \int_0^4 x \sqrt{x^2 + 1}\, dx$.