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

The integral of (1+x)^3 is (1/4)(1+x)^4 + C, where C is the constant of integration.

To find the integral of (1+x)^3, we can use the power rule of integration. This states that the integral of x^n is (1/(n+1))x^(n+1) + C, where C is the constant of integration. Applying this rule to (1+x)^3, we get:

∫(1+x)^3 dx = (1/4)(1+x)^4 + C

To check our answer, we can differentiate it using the power rule of differentiation. This states that the derivative of x^n is nx^(n-1). Applying this rule to (1/4)(1+x)^4, we get:

d/dx [(1/4)(1+x)^4] = (1/4) * 4(1+x)^3 * 1 = (1+x)^3

This is the same as the integrand, so our answer is correct.

In general, when integrating a polynomial function, we can use the power rule of integration to find the antiderivative. This involves adding 1 to the power of x and dividing by the new power, then adding the constant of integration. For example, the integral of x^2 is (1/3)x^3 + C, and the integral of 2x^3 - 5x^2 + 4x - 3 is (1/2)x^4 - (5/3)x^3 + 2x^2 - 3x + C.