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Integration Examples

Q:

 ∫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:

∫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 have:  ∫14*u6 du

Using the power rule:

∫14*u6 du = 14/7 * u7 + C = 2u7 + C

And since u = (12x – 2x2), the final answer is:

(12x – 2x2)7 + C

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