IB Maths AI 5.5 Notes

Integration extends beyond simple polynomials. In this section, you work with indefinite integrals of $x^n$, $\sin x$, $\cos x$, $\frac{1}{x}$ and $e^x$, together with linear composites such as $f(ax+b)$, and integrals that can be solved by inspection or substitution.

An indefinite integral gives a family of antiderivatives, so every answer must include a constant of integration $C$.

$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$

• $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$, where $n\in\mathbb{Q}$ and $n\ne-1$
• $\int \sin x\,dx=-\cos x+C$
• $\int \cos x\,dx=\sin x+C$
• $\int e^x\,dx=e^x+C$
• $\int \frac{1}{x}\,dx=\ln|x|+C$

The logarithm result is specifically required in the form $\ln|x|+C$.

Some examples include:

• $\int 6x^4\,dx=6\int x^4\,dx=6\left(\frac{x^5}{5}\right)+C=\frac{6}{5}x^5+C$
• $\int 2x^{-\frac{5}{3}}\,dx=2\left(\frac{x^{-\frac{2}{3}}}{-\frac{2}{3}}\right)+C=-3x^{-\frac{2}{3}}+C$
• $\int \left(x^4+2x^{-\frac{5}{3}}\right)\,dx=\frac{x^5}{5}-3x^{-\frac{2}{3}}+C$