7.1. Integration by Substitution http://www.ck12.org
∫ 1
cos^23 xdx=∫
sec^23 xdx.Substituting for the argument of the secant,u= 3 x,thendu= 3 dx,ordx=du/ 3 .Thus our integral becomes,
∫
sec^2 u.^13 du=^13∫
sec^2 udu
=^13 tanu+C
=^13 tan( 3 x)+C.Some integrations of trigonometric functions involve the logarithmic functions as a solution, as shown in the
following example.
Example 5:
Evaluate∫tanxdx.
Solution:
As you may have guessed, this is not a straightforward integration. We need to make use of trigonometric identities
to simplify it. Since tanx=sinx/cosx,
∫
tanxdx=∫ sinx
cosxdx.Now make a change of variablex.Chooseu=cosx.Thendu=−sinxdx,ordx=−du/sinx.Substituting,
∫ sinx
cosxdx=∫ sinx
u(−du
sinx)
=−
∫ du
u.This integral should look obvious to you. The integrand is the derivative of the natural logarithm lnu.
=−ln|u|+C
=−ln|cosx|+C.Another way of writing it, since−ln|u|=ln|^1 u|, is
=ln∣∣
∣∣^1
cosx∣∣
∣∣+C
=ln|secx|+C.