Reverse Chain Rule and Substitution

Look for an inner function together with a constant multiple of its derivative. SL5.10 permits either inspection or a written substitution.

\(\int k\mathrm{g}'(x)\mathrm{f}(\mathrm{g}(x))\,\mathrm{d}x=k\int\mathrm{f}(u)\,\mathrm{d}u,\qquad u=\mathrm{g}(x)\)

The power form requires \(n\ne-1\). The logarithmic form requires \(\mathrm{g}(x)\ne0\) throughout the working interval.

Write the substitution and its differential. Rewrite the entire integral in the new variable. Integrate, then return to the original variable for an indefinite integral.

\(\begin{aligned}u&=x^2+1,\qquad\mathrm{d}u=2x\,\mathrm{d}x\\ \int x(x^2+1)^{10}\,\mathrm{d}x&=\dfrac{1}{2}\int u^{10}\,\mathrm{d}u\\&=\dfrac{(x^2+1)^{11}}{22}+C\end{aligned}\)

For a definite integral, change both limits to the new variable and keep them in their transformed order. Never combine new-variable limits with an old-variable antiderivative.