7.1. Integration by Substitution http://www.ck12.org
Example 1:
Evaluate∫(x+ 1 )^5 dx.
Solution:
Letu=x+ 1 .Thendu=d(x+ 1 ) = 1 dx=dx.Substituting foruandduwe get
∫
(x+ 1 )^5 dx=∫
u^5 du.Integrating using the power rule,
=u6
6 +C.Sinceu=x+ 1 ,substituting back,
=(x+^1 )6
6 +C.Example 2:
Evaluate∫√ 4 x+ 3 dx.
Solution:
Letu= 4 x+ 3 .Thendu= 4 dx.Solving fordx,
dx=du/ 4.Substituting,
=
∫
u^1 /^2 ·^14 dx
=^14∫
u^1 /^2 dx=^14 u3 / 2
3 / 2 +C.Simplifying,
=^16 u^3 /^2 +C
=^16 ( 4 x+ 3 )^3 /^2 +C.