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