Definite Integrals — Practice Problems and Worked Solutions

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16 worked problems

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

1. 1.

   Evaluate the definite integral.

   $${\int }_0^{2}2xdx$$

   Answer: $4$

   Step-by-step solution

   1. $$\text{Antiderivative: }{x}^2$$
   2. $$\text{Evaluate }{x}^2{|}_0^2=4-0=4$$
2. 2.

   Evaluate the definite integral.

   $${\int }_0^{1}3x^{2}dx$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{Antiderivative: }{x}^3$$
   2. $$\text{Evaluate }{x}^3{|}_0^1=1-0=1$$
3. 3.

   Evaluate the definite integral.

   $${\int }_1^{3}xdx$$

   Answer: $4$

   Step-by-step solution

   1. $$\text{Antiderivative: }\frac{x^2}{2}$$
   2. $$\text{Evaluate }\frac{x^2}{2}{|}_1^3=\frac{9}{2}-\frac{1}{2}=4$$
4. 4.

   Evaluate the definite integral.

   $${\int }_0^{1}4x^{3}dx$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{Antiderivative: }{x}^4$$
   2. $$\text{Evaluate }{x}^4{|}_0^1=1-0=1$$
5. 5.

   Evaluate the definite integral.

   $${\int }_0^{4}xdx$$

   Answer: $8$

   Step-by-step solution

   1. $$\text{Antiderivative: }\frac{x^2}{2}$$
   2. $$\text{Evaluate }\frac{x^2}{2}{|}_0^4=8-0=8$$
6. 6.

   Evaluate the definite integral.

   $${\int }_0^{2}3xdx$$

   Answer: $6$

   Step-by-step solution

   1. $$\text{Antiderivative: }\frac{3x^2}{2}$$
   2. $$\text{Evaluate }\frac{3x^2}{2}{|}_0^2=6-0=6$$
7. 7.

   Evaluate the definite integral.

   $${\int }_0^3{x}^{2}dx$$

   Answer: $9$

   Step-by-step solution

   1. $$\text{Antiderivative: }\frac{x^3}{3}$$
   2. $$\text{Evaluate }\frac{x^3}{3}{|}_0^3=9-0=9$$
8. 8.

   Evaluate the definite integral.

   $${\int }_1^{2}6xdx$$

   Answer: $9$

   Step-by-step solution

   1. $$\text{Antiderivative: }3x^2$$
   2. $$\text{Evaluate }3x^2{|}_1^2=12-3=9$$
9. 9.

   Evaluate the definite integral.

   $${\int }_0^{3}2x^{2}dx$$

   Answer: $18$

   Step-by-step solution

   1. $$\text{Antiderivative: }\frac{2x^3}{3}$$
   2. $$\text{Evaluate }\frac{2x^3}{3}{|}_0^3=18-0=18$$
10. 10.

    Evaluate the definite integral.

    $${\int }_0^2{x}^{3}dx$$

    Answer: $4$

    Step-by-step solution

    1. $$\text{Antiderivative: }\frac{x^4}{4}$$
    2. $$\text{Evaluate }\frac{x^4}{4}{|}_0^2=4-0=4$$
11. 11.

    Evaluate the definite integral.

    $${\int }_2^{4}5xdx$$

    Answer: $30$

    Step-by-step solution

    1. $$\text{Antiderivative: }\frac{5x^2}{2}$$
    2. $$\text{Evaluate }\frac{5x^2}{2}{|}_2^4=40-10=30$$
12. 12.

    Evaluate the definite integral.

    $${\int }_0^{3}4xdx$$

    Answer: $18$

    Step-by-step solution

    1. $$\text{Antiderivative: }2x^2$$
    2. $$\text{Evaluate }2x^2{|}_0^3=18-0=18$$
13. 13.

    Evaluate the definite integral.

    $${\int }_0^2{x}^2+1dx$$

    Answer: $\frac{14}{3}$

    Step-by-step solution

    1. $$\text{Antiderivative: }\frac{x^3}{3}+x$$
    2. $$\text{Evaluate }\frac{x^3}{3}+x{|}_0^2=\frac{14}{3}-0=\frac{14}{3}$$
14. 14.

    Evaluate the definite integral.

    $${\int }_1^{3}2x+1dx$$

    Answer: $10$

    Step-by-step solution

    1. $$\text{Antiderivative: }{x}^2+x$$
    2. $$\text{Evaluate }{x}^2+x{|}_1^3=12-2=10$$
15. 15.

    Evaluate the definite integral.

    $${\int }_0^{2}6x^{2}dx$$

    Answer: $16$

    Step-by-step solution

    1. $$\text{Antiderivative: }2x^3$$
    2. $$\text{Evaluate }2x^3{|}_0^2=16-0=16$$
16. 16.

    Evaluate the definite integral.

    $${\int }_0^2{x}^3+xdx$$

    Answer: $6$

    Step-by-step solution

    1. $$\text{Antiderivative: }\frac{x^4}{4}+\frac{x^2}{2}$$
    2. $$\text{Evaluate }\frac{x^4}{4}+\frac{x^2}{2}{|}_0^2=6-0=6$$

More from Calculus I

• Evaluating Limits
• One-Sided Limits
• 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
• Antiderivatives
• Fundamental Theorem of Calculus
• u-Substitution
• Area Between Curves
• Volumes of Revolution