Integration formula

If you remember back to the power rule for differentiation, the derivative of ax^n is:

a\times n\times x^{n-1}

We can rearrange this to find the inverse, which gives our formula for integration:

\int ax^n\space dx = a\div(n+1) \times x^{n+1}

We use c here because we don’t know what the constant term on the end was. This constant was removed during differentiation - we can’t know what it will be after we integrate.

Find \int 4x+3 \space dx

• \int ax^n\space dx=a\div(n+1) \times x^{n+1}:
  • \int4x\space dx=4\div(1+1)\times x^{1+1}=2x^2
  • \int3\space dx=3\div(0+1)\times x^{0+1}3x
• Answer: 2x^2+3x+c

Find the integral of \frac{dy}{dx}=9x-4

• \int ax^n\space dx=a\div(n+1) \times x^{n+1}:
  • \int9x^1\space dx=9\div(1+1)\times x^{1+1}=\frac92x^2
  • \int4x^0\space dx=4\div(0+1)\times x^{0+1}=4x^1=4x
• Answer: \frac92x^2+4x+c

Given that f'(x)=\sqrt x-4x^{-\frac34} and that f(0)=3, find f(x)

• \int ax^n\space dx=a\div(n+1) \times x^{n+1}:
  • \int 1x^{\frac12}\space dx=1\div(\frac12+1)\times x^{\frac12+1}=\frac23x^{\frac32}
  • \int 4x^{-\frac34}\space dx=1\div(-\frac34+1)\times x^{-\frac34+1}=-\frac47x^{\frac14}

TODO: finish and correct example