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Section Integrals of Trigonometric Functions

When integrating trigonometric functions you will often use a combination of substitution, integration by parts, and trigonometric identities. Appendix Trigonometry  has the trigonometric identities that will be necessary for this section: the Pythagorean, double angle, half angle, and product identities.

Problem 5.19.

Evaluate \(\dsp \int \cos^3(x) \; dx\) by using the fact that \(\cos^{3} = \cos^2 \cdot \cos\) and the Pythagorean identity for \(\cos^2.\)

Problem 5.20.

Evaluate \(\dsp \int \cos^5 (x) \; dx\) by using the fact that \(\cos^5= \cos^4 \cdot \cos\)

Problem 5.21.

Evaluate \(\dsp \int \cos^{2n+1} (x) \; dx\) and \(\dsp \int \sin^{2n+1} (x) \; dx\) where \(n\) is a positive integer.

Problem 5.22.

Evaluate \(\dsp \int \sin^4 (x) \; dx\) using the half-angle identity for \(\sin.\)

Problem 5.23.

Evaluate each of the following integrals.

  1. \(\dsp \int \sin^3 (x) \cos^2(x) \; dx\)

  2. \(\dsp \int \sin^3 (x) \cos^5 (x) \; dx\)

  3. \(\dsp \int \sin^4(x) \cos^2 (x) \; dx\)

  4. \(\dsp \int \sin(3x) \cos(6x) \; dx\)

Problem 5.24.

Evaluate \(\dsp{ \int \csc(x) \; dx }\) by first multiplying by \(\dsp{ \frac{\csc(x) - \cot(x)} {\csc(x) - \cot(x)} }.\)

Problem 5.25.

Evaluate \(\dsp \int \tan^3(x) \sec^4(x) \; dx\) in the two different ways indicated.

  1. Use \(\tan^3 \cdot \sec^4 = \tan^2 \cdot \sec^3 \cdot \tan \cdot \sec\) and the Pythagorean identity for \(\tan^2.\)

  2. Use \(\tan^3 \cdot \sec^4 = \tan^3 \cdot \sec^2 \cdot \sec^2\) and the Pythagorean identity for \(\sec^2.\)

Problem 5.26.

Evaluate each of the following integrals.

  1. \(\dsp \int \cot^5(x) \; dx\)

  2. \(\dsp \int_{\frac{3\pi}{4}}^{\frac{5\pi}{4}} \sec^4(x) \; dx\)

  3. \(\dsp \int \cot^3(x) \csc^5(x) \; dx\)

  4. \(\dsp \int \tan^2(x) \sec^4(x) \; dx\)

  5. \(\dsp \int \tan^4(x) \sec^3(x) \; dx\)

Problem 5.27.

Not for presentation. Integrate the first, second, third, and fourth power of each of the six trigonometric functions. It is a well-documented fact that one of these (24) problems has appeared on at least one test in every Calculus II course since the days of Newton and Leibnitz.

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