Q:
Integrate: 14*(12 – 4x)(12x – 2x2)6 dx
Answer:
Since I see that (12 – 4x) is the derivative of (12x – 2x2), my instinct tells me to use u-substitution.
So, let
u = 12x – 2x2
then, compute du:
du = 12 – 4x dx
Now, we can directly substitute in u and du into our problem:
Integrate: 14*(12 – 4x)(12x – 2x2)6 dx = 14*u6 du
[the u replaced (12x – 2x2) and the du replaced the (12 – 4x) dx]
So, we are integrating: 14*u6 du
Using the power rule:
int[ 14*u6 du] = 14/7 * u7 + C = 2u7 + C
And since u = (12x – 2x2), the final answer is:
(12x – 2x2)7 + C