Trigonometric 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 trigonometric integral.

   $$\int {\sin }^2\left(x\right)dx$$

   Answer: $\frac{x}{2}-\frac{\sin \left(2x\right)}{4}+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int {\sin }^2\left(x\right)dx=\frac{x}{2}-\frac{\sin \left(x\right)\cos \left(x\right)}{2}+C$$
2. 2.

   Evaluate the trigonometric integral.

   $$\int {\cos }^2\left(x\right)dx$$

   Answer: $\frac{x}{2}+\frac{\sin \left(2x\right)}{4}+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int {\cos }^2\left(x\right)dx=\frac{x}{2}+\frac{\sin \left(x\right)\cos \left(x\right)}{2}+C$$
3. 3.

   Evaluate the trigonometric integral.

   $$\int \sin \left(x\right)\cos \left(x\right)dx$$

   Answer: $\frac{{\sin }^2\left(x\right)}{2}+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int \sin \left(x\right)\cos \left(x\right)dx=\frac{{\sin }^2\left(x\right)}{2}+C$$
4. 4.

   Evaluate the trigonometric integral.

   $$\int {\sec }^2\left(x\right)dx$$

   Answer: $\tan \left(x\right)+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int {\sec }^2\left(x\right)dx=\frac{\sin \left(x\right)}{\cos \left(x\right)}+C$$
5. 5.

   Evaluate the trigonometric integral.

   $$\int {\sin }^3\left(x\right)\cos \left(x\right)dx$$

   Answer: $\frac{{\sin }^4\left(x\right)}{4}+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int {\sin }^3\left(x\right)\cos \left(x\right)dx=\frac{{\sin }^4\left(x\right)}{4}+C$$
6. 6.

   Evaluate the trigonometric integral.

   $$\int \sin \left(x\right){\cos }^3\left(x\right)dx$$

   Answer: $-\frac{{\cos }^4\left(x\right)}{4}+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int \sin \left(x\right){\cos }^3\left(x\right)dx=-\frac{{\cos }^4\left(x\right)}{4}+C$$
7. 7.

   Evaluate the trigonometric integral.

   $$\int \sin \left(2x\right)dx$$

   Answer: $-\frac{\cos \left(2x\right)}{2}+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int \sin \left(2x\right)dx=-\frac{\cos \left(2x\right)}{2}+C$$
8. 8.

   Evaluate the trigonometric integral.

   $$\int \cos \left(3x\right)dx$$

   Answer: $\frac{\sin \left(3x\right)}{3}+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int \cos \left(3x\right)dx=\frac{\sin \left(3x\right)}{3}+C$$
9. 9.

   Evaluate the trigonometric integral.

   $$\int {\sin }^2\left(x\right)\cos \left(x\right)dx$$

   Answer: $\frac{{\sin }^3\left(x\right)}{3}+C$

   Step-by-step solution

   1. $$\text{Use a power-reduction identity or save one odd factor.}$$
   2. $$\int {\sin }^2\left(x\right)\cos \left(x\right)dx=\frac{{\sin }^3\left(x\right)}{3}+C$$
10. 10.

    Evaluate the trigonometric integral.

    $$\int \sin \left(x\right){\cos }^2\left(x\right)dx$$

    Answer: $-\frac{{\cos }^3\left(x\right)}{3}+C$

    Step-by-step solution

    1. $$\text{Use a power-reduction identity or save one odd factor.}$$
    2. $$\int \sin \left(x\right){\cos }^2\left(x\right)dx=-\frac{{\cos }^3\left(x\right)}{3}+C$$
11. 11.

    Evaluate the trigonometric integral.

    $$\int {\sin }^4\left(x\right)\cos \left(x\right)dx$$

    Answer: $\frac{{\sin }^5\left(x\right)}{5}+C$

    Step-by-step solution

    1. $$\text{Use a power-reduction identity or save one odd factor.}$$
    2. $$\int {\sin }^4\left(x\right)\cos \left(x\right)dx=\frac{{\sin }^5\left(x\right)}{5}+C$$
12. 12.

    Evaluate the trigonometric integral.

    $$\int {\tan }^2\left(x\right)dx$$

    Answer: $-x+\tan \left(x\right)+C$

    Step-by-step solution

    1. $$\text{Use a power-reduction identity or save one odd factor.}$$
    2. $$\int {\tan }^2\left(x\right)dx=-x+\frac{\sin \left(x\right)}{\cos \left(x\right)}+C$$
13. 13.

    Evaluate the trigonometric integral.

    $$\int \sin \left(3x\right)\cos \left(3x\right)dx$$

    Answer: $\frac{{\sin }^2\left(3x\right)}{6}+C$

    Step-by-step solution

    1. $$\text{Use a power-reduction identity or save one odd factor.}$$
    2. $$\int \sin \left(3x\right)\cos \left(3x\right)dx=\frac{{\sin }^2\left(3x\right)}{6}+C$$
14. 14.

    Evaluate the trigonometric integral.

    $$\int \sin \left(x\right){\cos }^4\left(x\right)dx$$

    Answer: $-\frac{{\cos }^5\left(x\right)}{5}+C$

    Step-by-step solution

    1. $$\text{Use a power-reduction identity or save one odd factor.}$$
    2. $$\int \sin \left(x\right){\cos }^4\left(x\right)dx=-\frac{{\cos }^5\left(x\right)}{5}+C$$
15. 15.

    Evaluate the trigonometric integral.

    $$\int {\sin }^2\left(x\right){\cos }^3\left(x\right)dx$$

    Answer: $-\frac{{\sin }^5\left(x\right)}{5}+\frac{{\sin }^3\left(x\right)}{3}+C$

    Step-by-step solution

    1. $$\text{Use a power-reduction identity or save one odd factor.}$$
    2. $$\int {\sin }^2\left(x\right){\cos }^3\left(x\right)dx=-\frac{{\sin }^5\left(x\right)}{5}+\frac{{\sin }^3\left(x\right)}{3}+C$$
16. 16.

    Evaluate the trigonometric integral.

    $$\int {\sec }^2\left(2x\right)dx$$

    Answer: $\frac{\tan \left(2x\right)}{2}+C$

    Step-by-step solution

    1. $$\text{Use a power-reduction identity or save one odd factor.}$$
    2. $$\int {\sec }^2\left(2x\right)dx=\frac{\sin \left(2x\right)}{2\cos \left(2x\right)}+C$$

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