Indefinite Integrals0%

Integration · Topic 2 of 9

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.

Theory

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

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

Where CC is the constant of integration.

⚠️ Common Examiner Traps

  • The + C+\,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 dxdx: integral notation must be accurate. An integral sign with no dxdx is incomplete.
  • Deal with the coefficient: the coefficient of xx is easily mishandled. Integrating (ax+b)n(ax+b)^n needs the extra 1a\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−1x^{-1} with the power rule: the rule fails when n=−1n = -1 because it would divide by zero.

Worked examples

Example 1

Find ∫x2 dx\int x^2 \,dx

∫x2 dx=x33+C\int x^2 \,dx = \frac{x^3}{3} + C

Example 2

Find ∫x−3 dx\int x^{-3} \,dx

∫x−3 dx=x−2−2+C=−12x2+C\int x^{-3} \,dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C

Example 3

Find ∫x5/4 dx\int x^{5/4} \,dx

∫x5/4 dx=x9/49/4+C=49x9/4+C\int x^{5/4} \,dx = \frac{x^{9/4}}{9/4} + C = \frac{4}{9}x^{9/4} + C

Example 4

Find ∫drr4\int \frac{dr}{r^4}

∫r−4 dr=r−3−3+C=−13r3+C\int r^{-4} \,dr = \frac{r^{-3}}{-3} + C = -\frac{1}{3r^3} + C

Example 5

Find ∫x54 dx\int \sqrt[4]{x^5} \,dx

∫x5/4 dx=x9/49/4+C=49x9/4+C\int x^{5/4} \,dx = \frac{x^{9/4}}{9/4} + C = \frac{4}{9}x^{9/4} + C

Example 6

Find ∫4x−3/2 dx\int 4x^{-3/2} \,dx

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

Example 7

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

4x3/83/8+3x22+7x+C=323x3/8+3x22+7x+C\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 ∫t(t+3) dt\int t(t+3) \,dt

First expand the brackets:

∫(t2+3t) dt=t33+3t22+C\int (t^2 + 3t) \,dt = \frac{t^3}{3} + \frac{3t^2}{2} + C

Example 9

Find ∫4m2−m+7m dm\int \frac{4m^2 - m + 7}{\sqrt{m}} \,dm

Split into separate fractions and simplify powers:

∫(4m3/2−m1/2+7m−1/2) dm\int \left( 4m^{3/2} - m^{1/2} + 7m^{-1/2} \right) \,dm
=4m5/25/2−m3/23/2+7m1/21/2+C= \frac{4m^{5/2}}{5/2} - \frac{m^{3/2}}{3/2} + \frac{7m^{1/2}}{1/2} + C
=85m5/2−23m3/2+14m1/2+C= \frac{8}{5}m^{5/2} - \frac{2}{3}m^{3/2} + 14m^{1/2} + C