**the question is “evaluate the integral exactly by using the fundamental theorem of calculus”?**

*
*

*I understand the fundamental theorem of calculus, but whenever i work a problem I get caught up on taking the antiderivatives… the equation that i need help finding the antiderivative of is*

(1+3x)^2

a step by step solution of this would be greatly apreciated

Well, I’m not an incredible calculus student myself, and there may be a faster way to do this, but I would start by actually multiplying out the equation.

Now, we need the antiderivative of: 1 + 6x + 9x^2

This is a lot easier. Let’s take one term at a time.

A) The way we learned it, you can find the antiderivative of x to a power with the formula “(x^(n + 1)) / (n+1)” (that’s “x to the n +1, all divided by n + 1)

So, 9x ^2 becomes 9[(x^3)/3], which equals 9 * (1/3) * x^3, or 3x^3. (Finding the derivative of this reveals that it is correct).

B) Using the same process, we know that the antiderivated of 6x is 3x^2.

C) The antiderivative of 1 is, of course, x.

So, our integral turns out to equal “3x^3 + 3x^2 + x + C”

**2005 AP Calculs AB Free Response Question**

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