**Q:**

**Integrate: 14*(12 – 4x)(12x – 2x ^{2})^{6} dx**

Answer:

Since I see that (12 – 4x) is the derivative of (12x – 2x^{2}), my instinct tells me to use u-substitution.

So, let

u = 12x – 2x^{2}

then, compute du:

du = 12 – 4x dx

Now, we can directly substitute in u and du into our problem:

Integrate: 14*(12 – 4x)(12x – 2x^{2})^{6} dx = 14*u^{6} du

[the u replaced (12x – 2x^{2}) and the du replaced the (12 – 4x) dx]

So, we are integrating: 14*u^{6} du

Using the power rule:

int[ 14*u^{6} du] = 14/7 * u^{7} + C = 2u^{7} + C

And since u = (12x – 2x^{2}), the final answer is:

(12x – 2x^{2})^{7} + C