Standard Integrals and Linear Composites

If \(\mathrm{F}^{\prime}(x)=\mathrm{f}(x)\), then a linear inner function contributes a constant scale factor:

\(\int\mathrm{f}(ax+b)\,\mathrm{d}x=\dfrac{1}{a}\mathrm{F}(ax+b)+C\qquad(a\ne0)\)

In the power rule, \(n\in\mathbb Q\) and \(n\ne-1\); choose a real interval on which the expression is defined. The reciprocal form requires \(ax+b\ne0\).

If \(\mathrm{F}^{\prime}(x)=2x\) and \(\mathrm{F}(1)=5\), then \(\mathrm{F}(x)=x^2+C\), \(C=4\), and \(\mathrm{F}(x)=x^2+4\).

Dividing by the inner derivative as a constant is valid only when the inner function is linear. For a nonlinear inner function, look for its derivative as a factor.

For example, \(\dfrac{\mathrm{d}}{\mathrm{d}x}(x^3+2)=3x^2\), while \(\int3x^2\,\mathrm{d}x=x^3+C\). Different constants give the same derivative.

For squared sine or cosine, first use \(\sin^2x=\dfrac{1-\cos2x}{2}\) or \(\cos^2x=\dfrac{1+\cos2x}{2}\). Then integrate each term, retaining the linear scale factor.