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Find the antiderivative of

f(x) = 3x^2

Dec 12, 2023

Find the antiderivative of

f(x) = 3x^2

Solution by Steps

step 1

To find the antiderivative of

f(x) = 3x^2

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

x^n

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

, where

n \neq -1

step 2

Applying the power rule to

3x^2

, we increase the exponent by 1 to get

x^3

and then divide by the new exponent, 3, to get

\frac{x^3}{3}

. Finally, we multiply by the coefficient, 3, from the original function

step 3

The antiderivative of

3x^2

3 \cdot \frac{x^3}{3}

. The 3s cancel out, leaving us with just

x^3

step 4

Don't forget to add the constant of integration,

C

, since the antiderivative is not unique

Answer

The antiderivative of

f(x) = 3x^2

x^3 + C

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}

plus a constant of integration,

C