IB Maths AA 5.10 Notes

Further indefinite integrals

This section extends integration to a wider range of functions. The main ideas are further indefinite integrals, integrals involving linear composites, partial fractions, substitution, and integration by parts.

Some key results are:

• $\int \tan x\,dx=-\ln|\cos x|+C$
• $\int \sec x\tan x\,dx=\sec x+C$
• $\int \sec^2 x\,dx=\tan x+C$
• $\int \cosec x\cot x\,dx=-\cosec x+C$
• $\int \cosec^2 x\,dx=-\cot x+C$
• $\int a^x\,dx=\frac{a^x}{\ln a}+C$
• $\int \frac{1}{x\ln a}\,dx=\log_a|x|+C$
• $\int \frac{1}{\sqrt{1-x^2}}\,dx=\arcsin x+C$
• $\int -\frac{1}{\sqrt{1-x^2}}\,dx=\arccos x+C$
• $\int \frac{1}{1+x^2}\,dx=\arctan x+C$

These come directly from reversing the derivative formulas.