Standard Integrals and Linear Composites

If F (x)=f(x), then a linear inner function contributes a constant scale factor: (ax+b),dx= 1aF(ax+b)+C (a 0) **Integrand** | **Antiderivative** (ax+b)n | (ax+b)n+1a(n+1)+C 1ax+b | 1aln|ax+b|+C eax+b | 1a eax+b+C sin(ax+b) | - 1acos(ax+b)+C cos(ax+b) | 1asin(ax+b)+C In the power rule, n in Q and n!= -1; choose a real interval on which the expression is defined. The reciprocal form requires ax+b 0. If F (x)=2x and F(1)=5, then F(x)=x2+C, C=4, and F(x)=x2+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.