Integration by Substitution

Starting point: what you already know

• The chain rule: $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$. For example $\frac{d}{dx}(x^2+1{)}^6=6(x^2+1{)}^5\cdot 2x$.
• Basic integrals: $\int u^{n}du=\frac{u^{n+1}}{n+1}+C$ for $n\ne -1$, $\int e^{u}du=e^u+C$, $\int \frac{1}{u}du=\ln |u|+C$, $\int \cos udu=\sin u+C$.
• Definite integrals: ${\int }_a^{b}f(x)dx=F(b)-F(a)$.
• Index laws: $\sqrt{u}=u^{1/2}$ and $\frac{1}{\sqrt{u}}=u^{-1/2}$.

The idea

Look at $\int 2x(x^2+1{)}^{5}dx$. The inside, $x^2+1$, has derivative $2x$, and $2x$ is sitting right there. That is the sign that substitution will work.

Definite integrals: change the limits

When $x$ goes from $a$ to $b$, work out where $u$ goes. Then you never have to switch back to $x$.

Quick summary

• Use substitution when an inside function and its derivative both appear.
• Choose $u$, find $du$, replace everything, integrate, then go back to $x$.
• For a definite integral, change the limits to $u$ values.
• Fix missing constants: if $du=3x^{2}dx$ then $x^{2}dx=\frac{1}{3}du$.
• Differentiate your answer to check it.

Quiz