EXAMPLE 1
 Use the General Power Rule to finding the following indefinite integrals.
∫ - 4 x ( 1 - 2 x 2 ) 2 d x

SOLUTION
 Letting u = 1 - 2 x 2 , we have d u / d x = - 4 x .

∫ - 4 x ( 1 - 2 x 2 ) 2 d x     = ∫ - 4 x ( 1 - 2 x 2 ) 2 d x = ∫ ( 1 - 2 x 2 ) - 2 ( - 4 x ) d x =
= ( 1 - 2 x 2 ) - 1 - 1 + C =
= - 1 1 - 2 x 2 + C


Multiplying And Dividing by a Constant

EXAMPLE 2
Find the indefinite integral
∫ x ( 3 - 4 x 2 ) 2 d x

SOLUTION
If we let u = 3 - 4 x 2 , it follows that du / dx = - 8x.
Now, we see that the factor -8 is not a part of integrand. However, we can adjust the integrand by multipluing and dividing by -8 as follows.
∫ x ( 3 - 4 x 2 ) 2 d x = ∫ - 1 8 ( 3 - 4 x 2 ) 2 ( - 8 x ) d x

Now, because - 1 8   is a constant, we can factor it out of the right-hand integral.
− 1 8     ∫ (3−4 x 2 ) 2 ⏞ u 2     (− 8 x) ⏞ du dx     d x

Finally, applying the General Power Rule produces
( - 1 8 ) ( 3 - 4 x 2   ) 3 3 + C = - ( 3 - 4 x 2   ) 3 24 + C

REMARK
Try using the Chain Rule, to check the result of this Example. After differentiating -   ( 3 - 4 x 2   ) 3 24     and simplyfying, you should obtain the original integrand.


In exercises 1-6, complete the table by identifying u and d u / d x   for given integral

∫ u n d u d x d x
u
d u d x
1.
 ∫ ( 5 x 2 + 1 ) 2 ( 10 x ) d x
2.
 ∫ ( 3 - 4 x 2 ) 2 ( - 8 x ) d x
3.
 ∫ 1 - x 2 ( - 2 x ) d x
4.
 ∫ 3 x 2 x 3 + 1 d x
5.
 ∫ ( 4 + 1 x 2 ) ( - 2 x 3 ) d x
6.
 ∫ 1 ( 1 + 2 x ) 2 ( 2 ) d x