Indefinite Integrals

Video lesson · from 0:179 worked examples

One lesson video covers all of Integration, so it opens at 0:17 for this topic — not from the beginning.

When integrating a term of the form $ax^n$ , we increase the power by 1 and divide by the new power:

\int ax^n \,dx = \frac{ax^{n+1}}{n+1} + C \quad (n \neq -1)

Where $C$ is the constant of integration.

The $+\,C$ is not optional: omitting the constant of integration invalidates the working that follows — it costs more than one mark because everything after it is wrong too.
Write $dx$ : integral notation must be accurate. An integral sign with no $dx$ is incomplete.
Deal with the coefficient: the coefficient of $x$ is easily mishandled. Integrating $(ax+b)^n$ needs the extra $\frac{1}{a}$ .
Rewrite first: roots and fractions must be written as indices before you can integrate. You cannot integrate a product or quotient term by term as it stands.
Never integrate $x^{-1}$ with the power rule: the rule fails when $n = -1$ because it would divide by zero.

Example 1

Find $\int x^2 \,dx$

\int x^2 \,dx = \frac{x^3}{3} + C

Example 2

Find $\int x^{-3} \,dx$

\int x^{-3} \,dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C

Example 3

Find $\int x^{5/4} \,dx$

\int x^{5/4} \,dx = \frac{x^{9/4}}{9/4} + C = \frac{4}{9}x^{9/4} + C

Example 4

Find $\int \frac{dr}{r^4}$

\int r^{-4} \,dr = \frac{r^{-3}}{-3} + C = -\frac{1}{3r^3} + C

Example 5

Find $\int \sqrt[4]{x^5} \,dx$

\int x^{5/4} \,dx = \frac{x^{9/4}}{9/4} + C = \frac{4}{9}x^{9/4} + C

Example 6

Find $\int 4x^{-3/2} \,dx$

\int 4x^{-3/2} \,dx = \frac{4x^{-1/2}}{-1/2} + C = -8x^{-1/2} + C = -\frac{8}{\sqrt{x}} + C

Example 7

Find $\int (4x^{-5/8} + 3x + 7) \,dx$

\frac{4x^{3/8}}{3/8} + \frac{3x^2}{2} + 7x + C = \frac{32}{3}x^{3/8} + \frac{3x^2}{2} + 7x + C

Example 8

Find $\int t(t+3) \,dt$

First expand the brackets:

\int (t^2 + 3t) \,dt = \frac{t^3}{3} + \frac{3t^2}{2} + C

Example 9

Find $\int \frac{4m^2 - m + 7}{\sqrt{m}} \,dm$

Split into separate fractions and simplify powers:

\int \left( 4m^{3/2} - m^{1/2} + 7m^{-1/2} \right) \,dm

= \frac{4m^{5/2}}{5/2} - \frac{m^{3/2}}{3/2} + \frac{7m^{1/2}}{1/2} + C

= \frac{8}{5}m^{5/2} - \frac{2}{3}m^{3/2} + 14m^{1/2} + C