Antiderivatives — Practice Problems and Worked Solutions

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18 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the antiderivative.

   $$\int 6x^{2}dx$$

   Answer: $2x^3+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 6x^{2}dx=\frac{6}{3}{x}^3+C$$
   3. $$=2x^3+C$$
2. 2.

   Find the antiderivative.

   $$\int 4x^{3}dx$$

   Answer: $x^4+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 4x^{3}dx=\frac{4}{4}{x}^4+C$$
   3. $$=x^4+C$$
3. 3.

   Find the antiderivative.

   $$\int 3xdx$$

   Answer: $\frac{3x^2}{2}+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 3xdx=\frac{3}{2}{x}^2+C$$
   3. $$=\frac{3x^2}{2}+C$$
4. 4.

   Find the antiderivative.

   $$\int 10x^{4}dx$$

   Answer: $2x^5+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 10x^{4}dx=\frac{10}{5}{x}^5+C$$
   3. $$=2x^5+C$$
5. 5.

   Find the antiderivative.

   $$\int 2x^{5}dx$$

   Answer: $\frac{x^6}{3}+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 2x^{5}dx=\frac{2}{6}{x}^6+C$$
   3. $$=\frac{x^6}{3}+C$$
6. 6.

   Find the antiderivative.

   $$\int 5dx$$

   Answer: $5x+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 5dx=\frac{5}{1}{x}^1+C$$
   3. $$=5x+C$$
7. 7.

   Find the antiderivative.

   $$\int 8x^{3}dx$$

   Answer: $2x^4+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 8x^{3}dx=\frac{8}{4}{x}^4+C$$
   3. $$=2x^4+C$$
8. 8.

   Find the antiderivative.

   $$\int 9x^{2}dx$$

   Answer: $3x^3+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 9x^{2}dx=\frac{9}{3}{x}^3+C$$
   3. $$=3x^3+C$$
9. 9.

   Find the antiderivative.

   $$\int 12x^{5}dx$$

   Answer: $2x^6+C$

   Step-by-step solution

   1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
   2. $$\int 12x^{5}dx=\frac{12}{6}{x}^6+C$$
   3. $$=2x^6+C$$
10. 10.

    Find the antiderivative.

    $$\int 7xdx$$

    Answer: $\frac{7x^2}{2}+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 7xdx=\frac{7}{2}{x}^2+C$$
    3. $$=\frac{7x^2}{2}+C$$
11. 11.

    Find the antiderivative.

    $$\int 6x^{4}dx$$

    Answer: $\frac{6x^5}{5}+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 6x^{4}dx=\frac{6}{5}{x}^5+C$$
    3. $$=\frac{6x^5}{5}+C$$
12. 12.

    Find the antiderivative.

    $$\int 4x^{6}dx$$

    Answer: $\frac{4x^7}{7}+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 4x^{6}dx=\frac{4}{7}{x}^7+C$$
    3. $$=\frac{4x^7}{7}+C$$
13. 13.

    Find the antiderivative.

    $$\int 15x^{2}dx$$

    Answer: $5x^3+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 15x^{2}dx=\frac{15}{3}{x}^3+C$$
    3. $$=5x^3+C$$
14. 14.

    Find the antiderivative.

    $$\int 2x^{7}dx$$

    Answer: $\frac{x^8}{4}+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 2x^{7}dx=\frac{2}{8}{x}^8+C$$
    3. $$=\frac{x^8}{4}+C$$
15. 15.

    Find the antiderivative.

    $$\int 10xdx$$

    Answer: $5x^2+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 10xdx=\frac{10}{2}{x}^2+C$$
    3. $$=5x^2+C$$
16. 16.

    Find the antiderivative.

    $$\int 3x^{4}dx$$

    Answer: $\frac{3x^5}{5}+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 3x^{4}dx=\frac{3}{5}{x}^5+C$$
    3. $$=\frac{3x^5}{5}+C$$
17. 17.

    Find the antiderivative.

    $$\int 14x^{6}dx$$

    Answer: $2x^7+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 14x^{6}dx=\frac{14}{7}{x}^7+C$$
    3. $$=2x^7+C$$
18. 18.

    Find the antiderivative.

    $$\int 8dx$$

    Answer: $8x+C$

    Step-by-step solution

    1. $$\text{Reverse the power rule: raise the exponent, divide by the new exponent.}$$
    2. $$\int 8dx=\frac{8}{1}{x}^1+C$$
    3. $$=8x+C$$

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• Limits at Infinity
• Continuity
• Derivative from First Principles
• Power Rule
• Trigonometric Derivatives
• Exponential & Log Derivatives
• Product Rule
• Quotient Rule
• Chain Rule
• Implicit Differentiation
• Related Rates
• Curve Analysis
• Optimization
• Definite Integrals
• Fundamental Theorem of Calculus
• u-Substitution
• Area Between Curves
• Volumes of Revolution