calculus math homework problem

Question

Suppose f(x) =x^2+sin x

and F is an antiderivative of f that satisfies f(pi ) = 0

then f(x) =?

Answer 1

I will assume your question reads: ". . . and F is an antiderivative of f that satisfies F(pi ) = 0, then F(x) =?"

The antiderivative of x² is:

because, when you take the derivative of it, you get x²

The antiderivative of sin(x) is:

because, when you take the derivative of it, you get sin(x)

And, we must add a constant as the third term because when we take the derivative of a constant, we get zero. So, so far that gives us an antiderivative of:

F(x) = (1/3)x³ - cos(x) + C

To determine the value of "C", we must use the data point (π, 0) and plug it into our F(x):

0 = (1/3)(π)³ - cos(π) + C

C = -1 - (1/3)π³

So F(x) = (1/3)x³ - cos(x) - 1 - (1/3)π³

Answer 2

F(x) = ∫ f(x)dx = ∫ (x² + sinx)dx = (1/3)x³ - cosx + C

Since F(π) = 0, π³ / 3 + 1 + C = 0. So, C = -1 - (1/3)π³.

F(x) = (1/3)x³ - cosx - 1 - (1/3)π³