Evaluation by Substitution

When the integrand is a composite function times (a constant multiple of) the derivative of the inner function, the substitution method simplifies it. In a definite integral there is one extra step compared with the indefinite case: the limits of integration must be changed to match the new variable, after which no back-substitution to $x$ is needed.

Method — definite integral by substitution:

1. Choose a substitution $u = g(x)$ so that $du = g'(x)\,dx$ appears (up to a constant) in the integrand.
2. Change the limits. Compute the new lower limit $u = g(a)$ and new upper limit $u = g(b)$.
3. Rewrite the whole integral in terms of $u$ and the new limits: $\displaystyle\int_{a}^{b} f(g(x))\,g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$.
4. Evaluate the new definite integral in $u$ directly — do not convert back to $x$; the changed limits already carry all the information.

Illustration. Evaluate $\displaystyle\int_{0}^{1} 2x\,(x^{2}+1)^{3}\,dx$. Put $u = x^{2}+1$, so $du = 2x\,dx$. New limits: when $x=0$, $u = 0^{2}+1 = 1$; when $x=1$, $u = 1^{2}+1 = 2$. The integral becomes

Two common inner-function patterns at this level:

• $u =$ (a polynomial), when its derivative appears as a factor — e.g. $u = x^{2}+1$ above, or $u = x^{3}$ with $du = 3x^{2}\,dx$.
• $u = \log x$, with $du = \dfrac{1}{x}\,dx$, whenever a $\dfrac{1}{x}$ factor multiplies a function of $\log x$.