First year:
(I)
Integration by substitution
The method
Remember that
 f ( x)dx  F ( x) , where
dF ( x)
 f ( x) so, in a way, integration
dx
is the “opposite” of differentiation.
Integration by substitution is in this case the “opposite” of the chain rule for
differentiation.
2x
dx .
Let’s have a look at an example: calculate  2
x 1
We are looking for a function F (x) such that
dF ( x)
2x
 2 .
dx
x 1
Now, if we set u ( x)  x 2  1 , we get
2x
u ' ( x)

x  1 u ( x)
2
and we know that only one type of function F can have a derivative of this
form: the logarithm
dF ( x) u ' ( x)

.
dx
u ( x)
Note that we used the chain rule to differentiate F .
F ( x)  ln( u ( x)) 
Hence
x
2x
dx  ln( u ( x))  ln( x 2  1).
1
2
Now, “integration by substitution” means that you change the variable in order
to transform the function inside the integral into a standard function. Let’s
have a look at the example above using this method:
Set u  x 2  1 . Then
du
 2x 
dx
du  2 xdx ,
from which we get
x
2x
1
dx   du  ln( u )  ln( x 2  1).
u
2
2
(II)
Worked example and exercises
(i)
Calculate
 xx
2
 1 dx.
6
Set u  x 2  1 . Then
1
du  xdx , so that
2


7
1 6
1 u7
x2 1
x
x

1
dx

u
du


.

2
2 7
14

2

6
(ii)
Calculate  sin( x)cos( x)  3 dx.
(iii)
Calculate
x2
 x 3  1 dx .
(iv)
Calculate
ex 1
 2e x  2 x dx.
(v)
Calculate
 sin( 2 x)  3 dx.
(vi)
x
(vii)
5
2

x
ln( x 2  1)  2
1
x 2e x
e
cos( 2 x)
x3
3
1
dx.

3
dx.