Find antiderivative of f(x) = x?

Question

Find the antiderviative of f(x) = x

Answer is: 1/2x^2 + C

I understand the constant part of it but the 1/2X^2?

Answer 1

Differentiate 1/2 x² + C:

d(1/2x² + C)/dx = 1/2 d(x²)/dx + d(C)/dx = 1/2 * 2x + 0 = x

The 1/2x² is the important part, it's the part that gives you the derivative of x. The constant part just generalizes it so that you have _all_ the antiderivatives of x, not just one.

Answer 2

to find anti-derivative of f(x)
you add 1 to the power
divide by the new power
add a constant

in this case, the power of x is 1. Add 1 to get a power of 2... x^2

and then divide by the power, 2 ... same as 1/2

add constant c

Answer 1/2x^2 + c

Hope that helps

Answer 3

Use the power rule: int(x^n dx) = 1/(n+1) x^(n+1). So in this case, n=1, thus that is were you get 1/2 x^2 + c

Answer 4

an antiderivative is simply an integral. To find an integral the equation is:

[x/(n+1)]^(n+1)

where n is the order/power of your variable.

for instance use x^1

the integral is (x/2)^2

because the n=1, you simply plug it into the equation.