This page covers Integration techniques.
Rewriting the Integrand
If you are asked to integrate a fraction, try multiplying or dividing the top and bottom of the fraction by a number.
If we divide everything on the numerator and everything on the denominator by x2, we get:
= ∫ (3x – 4x-1 – 5x-2) dx
= 3x2 – 4lnx + 5x-1 + c
Split into Partial Fractions
Sometimes it will help if you split a fraction up before attempting to integrate. This can be done using the method of partial fractions.
(Here we split the fraction into partial fractions)
= -3lnx + 4ln(x - 1) + (x - 1)-1 + c
Using Trigonometric Formulae
When integrating trigonometric expressions, it will often help to rewrite the integral using trigonometric formulae.
∫ cos2x dx
cos2x = 2cos2x - 1
cos2x = ½ (cos2x + 1)
∫ cos2x dx = ½ ∫ (cos2x + 1) dx
= ½ ( ½ sin2x + x) + c
= ¼ sin2x + ½ x + c