I need help with math integrals

1. Evaluate the integral. ∫ (2 − 5x) dx

2. Find f(x) if f(1) = −1 and the tangent line at (x, f(x)) has slope 8e^x + 6.

3. Decide on what substitution to use, and then evaluate the given integral using a substitution.
∫ (4x + 7)^3 dx

4. Decide on what substitution to use, and then evaluate the given integral using a substitution.
∫ 12x (−x^2 + 2) dx <---Here, what's in parentheses is actually the square root of -x^2+2

5. Decide on what substitution to use, and then evaluate the given integral using a substitution.
∫ x(x − 5)^1/5 dx

∫ (2 - 5x) dx
∫ 2 dx - ∫ 5x dx
2 ∫ dx - 5 ∫ x dx
2x - 5 (1/2 x²) + C
2x - 5/2 x² + C
- - - - - - - - - - - -

Given f '(x) = 8e^x + 6.

f(x) = ∫ f '(x) dx
f(x) = ∫ (8e^x + 6) dx
f(x) = 8 ∫ e^x dx + 6 ∫ dx
f(x) = 8e^x + 6x + C

Given that f(1) = -1,
-1 = 8e^(1) + 6(1) + C
-1 = 8e + 6 + C
-8e - 7 = C
-(8e + 7) = C

f(x) = 8e^x + 6x - (8e + 7)
- - - - - - - - - - - -

∫ (4x + 7)³ dx

Let u = 4x + 7
du = 4 dx
1/4 du = dx

1/4 ∫ u³ du
1/4 (1/4 u⁴) + C
1/16 u⁴ + C
1/16 (4x + 7)⁴ + C
- - - - - - - - - - - -

∫ 12x √(2 - x²) dx
6 ∫ 2x √(2 - x²) dx

Let u = 2 - x²
du = -2x dx
- du = 2x dx

-6 ∫ √u du
-6 (2/3 u^(3/2)) + C
-4 u^(3/2) + C
-4(2 - x²)^(3/2) + C
- - - - - - - - - - - -

∫ x(x - 5)^(1/5) dx

Let u = x - 5
du = dx
From this, you also have
x = u + 5

∫ (u + 5) u^(1/5) du
∫ (u^(6/5) + 5u^(1/5)) du
∫ u^(6/5) du + 5 ∫ u^(1/5) du
5/11 u^(11/5) + 5 (5/6 u^(6/5)) + C
5/11 (x - 5)^(11/5) + 25/6 (x - 5)^(6/5) + C

These last few steps aren't necessary, I personally don't like too many rational exponents.
5/66 (x - 5)^(6/5) ∙ [6(x - 5)^(5/5) + 5] + C
5/66 (x - 5)^(6/5) ∙ [6x - 30 + 5] + C
5/66 (x - 5)^(6/5) ∙ (6x - 25) + C