How to Find an Antiderivative

Step-by-step solution

1. Write the power rule for integrals.

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

   Why: The rule reverses the power rule for derivatives.

2. Apply the rule with n = 2.

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

   Why: The power rises to 3 and the coefficient divides by the new power.

3. Simplify.

   $$x^3 + C$$

   Why: The 3 on top and the 3 on the bottom cancel.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $\int x\,dx$

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   $\frac{x^2}{2} + C$

2. 2

   $\int 2x\,dx$

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   $x^2 + C$

3. 3

   $\int x^2\,dx$

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

4. 4

   $\int 4x^3\,dx$

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   $x^4 + C$

5. 5

   $\int 5\,dx$

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   $5x + C$