The intergel (3x^2 -4x+10)^6 (12x-8)

Question

It should not be a fourteen-degree polynomial but thats what i keep getting

Answer 1

∫ (3x² - 4x + 10)⁶(12x - 8)dx

Let u = 3x² - 4x + 10. Then, du = (6x - 4)dx. So, 2du = (12x - 8)dx.

So, we have ∫ u⁶(2du) = 2 ∫ u⁶du = (2/7)u⁷ + C = (2/7)(3x² - 4x + 10)⁷ + C

Answer 2

Given integral:

∫ (3x^2 - 4x + 10)^6 (12x - 8) dx

1. Substitute:

Let u = 3x^2 - 4x + 10.

2. Differentiate `u` with respect to `x`:

du/dx = 6x - 4.

Rearranging gives:

du = (6x - 4) dx.

3. Express `12x - 8` in terms of `du`:

Note that 12x - 8 = 2(6x - 4). So:

12x - 8 dx = 2 du.

4. Substitute these into the integral:

∫ (3x^2 - 4x + 10)^6 (12x - 8) dx

becomes:

∫ u^6 * 2 du.

5. Simplify and integrate:

2 ∫ u^6 du.

To integrate `u^6`, we get:

∫ u^6 du = u^7 / 7.

So:

2 * (u^7 / 7) = (2 / 7) * u^7.

6. Substitute`u` back:

u = 3x^2 - 4x + 10.

Therefore:

∫ (3x^2 - 4x + 10)^6 (12x - 8) dx = (2 / 7) * (3x^2 - 4x + 10)^7 + C.

Here, `C` is the constant of integration.

Answer 3

It is a 14th degree polynomial.

Plug it into this integral calculator to see it: https://www.integral-calculator.com/