Introduction
Integration by substitution simplifies integrals that are otherwise too difficult to solve.
Think of it as a reverse chain rule.
\begin{align*}
u &= g(x) \\\\
F(g(b)) - F(g(a)) &= \int_a^b f(g(x))g'(x)dx \\\\
F(g(b)) - F(g(a)) &= \int_{g(a)}^{g(b)} f(u)du \\\\
\int_a^b f(g(x))g'(x)dx &= \int_{g(a)}^{g(b)} f(u)du
\end{align*}
Examples
1.
\int x^3 e^{x^4} \, dx \\ \text{} \\
u = x^4 \quad\quad \frac{du}{dx}=4x^3 \quad\quad \frac{du}{4} = x^3dx \\ \text{} \\
\begin{align*}
\int x^3 e^{x^4} \, dx &= \int \frac{1}{4}e^{u} \, du \\\\
\int \frac{1}{4}e^{u} \, du &= \frac{1}{4}e^{u} + C \\\\
\int \frac{1}{4}e^{u} \, du &= \frac{1}{4}e^{x^4} + C \\\\
\int x^3 e^{x^4} \, dx &= \frac{1}{4}e^{x^4} + C
\end{align*}
2.
\int 5x\sin(3x^2) \, dx \\ \text{} \\
u = 3x^2 \quad\quad \frac{du}{dx}=6x \quad\quad \frac{du}{6} = xdx \\ \text{} \\
\begin{align*}
\int 5x\sin(3x^2) \, dx &= \int \frac{5}{6}\sin(u) \, du \\\\
\int \frac{5}{6}\sin(u) \, du &= -\frac{5}{6}\cos(u) + C \\\\
\int \frac{5}{6}\sin(u) \, du &= -\frac{5}{6}\cos(3x^2) + C \\\\
\int 5x\sin(3x^2) \, dx &= -\frac{5}{6}\cos(3x^2) + C
\end{align*}
3.
\int \frac{1}{x\ln(x)} \, dx \\ \text{} \\
u = \ln(x) \quad\quad \frac{du}{dx}=\frac{1}{x} \quad\quad du = \frac{1}{x}dx \\ \text{} \\
\begin{align*}
\int \frac{1}{x\ln(x)} \, dx &= \int \frac{1}{u} \, du \\\\
\int \frac{1}{u} \, du &= \ln|u| + C \\\\
\int \frac{1}{u} \, du &= \ln|\ln(x)| + C \\\\
\int \frac{1}{x\ln(x)} \, dx &= \ln|\ln(x)| + C
\end{align*}