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Find the antiderivative of

f(x) = 3x^2

Nov 21, 2023

Find the antiderivative of

f(x) = 3x^2

Here is a Step-by-Step solution: #step 1#

To find the antiderivative of a function, we need to determine the function whose derivative gives us the original function. In this case, we are looking for a function

F(x)

such that

F'(x) = 3x^2

#step 2#

The antiderivative of

x^n

where

n

is a real number and

n \neq -1

, is given by

\frac{x^{n+1}}{n+1} + C

, where

C

is the constant of integration.

#step 3#

Applying the antiderivative formula to

3x^2

, we increase the exponent by 1 to get

x^3

and then divide by the new exponent, which is 3. We also multiply by the coefficient, which is 3, and add the constant of integration

C

#step 4#

The antiderivative of

3x^2

is therefore:

F(x) = \frac{3x^{2+1}}{2+1} + C = x^3 + C

Answer

The antiderivative of

f(x) = 3x^2

F(x) = x^3 + C

, where

C

is the constant of integration.

Key Concept

Antiderivative of a Power Function

Explanation

The antiderivative of

x^n

\frac{x^{n+1}}{n+1} + C

, which applies to

f(x) = 3x^2

to find the antiderivative

F(x) = x^3 + C