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Find the antiderivative of

f(x) = 3x^2

Apr 28, 2024

Find the antiderivative of

f(x) = 3x^2

Generated Graph

Solution by Steps

step 1

To find the antiderivative of

f(x) = 3x^2

, we integrate the function with respect to

x

step 2

The integral of

3x^2

with respect to

x

\int 3x^2 \, dx

step 3

Applying the power rule for integration, which states that

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

for any real number

n \neq -1

, we get

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

step 4

Simplifying the expression, we have

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

step 5

The constant multiples can be cancelled out, resulting in

x^3 + C

, where

C

is the constant of integration

Answer

x^3 + C

Key Concept

Integration and the Power Rule

Explanation

The antiderivative of

3x^2

is found by applying the power rule for integration, which increases the exponent by 1 and divides by the new exponent, and then multiplying by the constant outside the integral.