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Find the antiderivative of

f(x) = 3x^2

Nov 11, 2023

Find the antiderivative of

f(x) = 3x^2

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

To find the antiderivative of the function

f(x) = 3x^2

, we will use the power rule for integration, which states that the antiderivative of

x^n

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

, where

n

is a real number and

n \neq -1

#step 2#

Applying the power rule for integration to

f(x) = 3x^2

, we increase the exponent by 1, giving us

x^{2+1}

, and then divide by the new exponent, which is 3. We also need to include the constant of integration, often denoted as

C

, because the antiderivative is not unique.

#step 3#

Performing the integration, we get:

\int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} + C = \frac{3x^3}{3} + C = x^3 + C

Answer

The antiderivative of

f(x) = 3x^2

x^3 + C

, where

C

is the constant of integration.

Key Concept

Power Rule for Integration

Explanation

The power rule for integration is used to find the antiderivative of a function in the form of

x^n

, resulting in

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

, where

C

is the constant of integration. In this case, the function

3x^2

was integrated to

x^3 + C