Indefinite Integral Rules (College Board AP® Calculus BC): Revision Note

When integrating sums or differences of terms

• the integral is simply the sum (or difference) of the integrals of the terms

• This may be expressed as $\int \left(f\left(x\right)\pm g\left(x\right)\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

  • E.g. $\int \left(x^2+\cos x\right)dx=\int x^{2}dx+\int \cos xdx=\frac{1}{3}{x}^3+\sin x+C$

  • We still only need a single constant of integration

• Note that products and quotients of terms cannot be integrated in this way

  • They may need to be expanded or simplified first

  • Or a more advanced integration technique may need to be used

    • See the 'Methods of Integration' study guide

When integrating a constant multiple of a term

• the term may be brought out in front of the integral as a multiplier

• This may be expressed as $\int kf\left(x\right)dx=k\int f\left(x\right)dx$

  • E.g. $\int 5e^{x}dx=5\int e^{x}dx=5e^x+C$

The rules for sums, differences and constant multiples can be combined

• If $p$ and $q$ are constants, then

  • $\int \left(pf\left(x\right)\pm qg\left(x\right)\right)dx=p\int f\left(x\right)dx\pm q\int g\left(x\right)dx$

  • This idea can be extended for any number of terms inside the integral

Sometimes expressions need to be simplified before you can integrate them

This may involve expanding brackets

• E.g. $\int {\left(x^2+2\right)}^{2}dx$

• Expand the brackets

  • ${\left(x^2+2\right)}^2=x^4+4x^2+4$

• Now integrate in the usual way

  • $\int {\left(x^2+2\right)}^{2}dx=\int \left(x^4+4x^2+4\right)dx=\frac{1}{5}{x}^5+\frac{4}{3}{x}^3+4x+C$

Or rearranging fractions (including using laws of exponents)

• E.g. $\int \frac{5x^3-3}{x^2}dx$

• Rearrange the fraction

  • $\frac{5x^3-3}{x^2}=\frac{5x^3}{x^2}-\frac{3}{x^2}=5x-\frac{3}{x^2}$

• Use laws of exponents

  • $5x-\frac{3}{x^2}=5x-3x^{-2}$

• Now integrate in the usual way

  • $\int \frac{5x^3-3}{x^2}dx=\int \left(5x-3x^{-2}\right)dx=\frac{5}{2}{x}^2+3x^{-1}+C$