What's the integral of (2x+1)^3?

The integral of (2x+1)^3 is (4/5)(2x+1)^5 + C.

To solve this integral, we can use the power rule of integration, which states that the integral of x^n is (1/(n+1))x^(n+1) + C. We can apply this rule to each term in the expansion of (2x+1)^3, then add the resulting integrals together.

(2x+1)^3 = (2x)^3 + 3(2x)^2(1) + 3(2x)(1)^2 + 1^3
= 8x^3 + 12x^2 + 6x + 1

Now we can integrate each term using the power rule:

∫8x^3 dx = (8/4)x^4 + C = 2x^4 + C
∫12x^2 dx = (12/3)x^3 + C = 4x^3 + C
∫6x dx = (6/2)x^2 + C = 3x^2 + C
∫1 dx = x + C

Putting these integrals back into the original equation, we get:

∫(2x+1)^3 dx = 2x^4 + 4x^3 + 3x^2 + x + C

However, we can simplify this further by factoring out a common factor of 2x+1:

∫(2x+1)^3 dx = ∫(2x+1)(2x+1)^2 dx
= (1/2)∫(2x+1)^2 d(2x+1)^2
= (1/2)(1/3)(2x+1)^3 + C
= (1/6)(2x+1)^3 + C

Finally, we can simplify this even further by multiplying through by 4/2:

∫(2x+1)^3 dx = (4/5)(2x+1)^5 + C

Therefore, the integral of (2x+1)^3 is (4/5)(2x+1)^5 + C.