Integration by substitution

When the integrand contains a composite function whose “inner derivative” appears (up to constants), it is convenient to change variable. The method is organised into five steps.

In brief — The 5 steps

1. Choose the substitution $t = g(x)$.
2. Compute the differential $dt = g'(x)\,dx$.
3. Obtain $dx$: $dx = \dfrac{dt}{g'(x)}$ and substitute.
4. Compute the antiderivative in $t$.
5. Inverse substitution $t = g(x)$.

Example — $\int_0^\pi x\sin(x^2)\,dx$

Set $t = x^2$, $dt = 2x\,dx$, $dx = dt/(2x)$: $\int_0^\pi x\sin(t)\cdot\frac{dt}{2x} = \frac{1}{2}\int\sin(t)\,dt = \frac{1}{2}\bigl[-\cos(t)\bigr] = \frac{1}{2}\bigl[-\cos(x^2)\bigr]_0^\pi = \frac{1}{2}\bigl[1-\cos(\pi^2)\bigr].$

Example — $\int_0^1\dfrac{x}{1+x^2}\,dx$

Set $t=1+x^2$, $dt=2x\,dx$: $\frac{1}{2}\int\frac{dt}{t} = \frac{1}{2}\ln|t| = \frac{1}{2}\bigl[\ln(1+x^2)\bigr]_0^1 = \frac{1}{2}\ln 2.$

The sign that the substitution works is that, after step 3, the variable $x$ disappears completely and there remains an integral in $t$ alone.

Links

Topics: Integrale
Concepts: Differenziale · Integrale per sostituzione
Methods: Integrale sostituzione
Skills: Calcolare · Integrare