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Integration Techniques, L’Hopital’s Rule, and Improper Integrals

8.3 Trigonometric Integrals

Focus on odd trigonometric powers, even powers and half-angle identities, mixed sine and cosine products.

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Ideas in this lesson

01Odd trigonometric powersWorked steps · five questions02Even powers and half-angle identitiesWorked steps · five questions03Mixed sine and cosine productsWorked steps · five questions
Idea 1 of 3

Odd trigonometric powers

Reserve one odd-power sine or cosine factor and substitute using the other function.

Worked example 1

Evaluate ∫0π2sin⁡2x dx\int_{0}^{\pi } 2\sin ^{2}x\,dx.
  1. sin⁡2x=1−cos⁡2x2\sin ^{2}x = \dfrac{1 - \cos 2x}{2}
  2. F(x)=22(x−sin⁡2x2)F(x) = \dfrac{2}{2}(x - \dfrac{\sin 2x}{2})
  3. F(π)−F(0)=2π2=π≈3.14F(\pi ) - F(0) = \dfrac{2\pi }{2} = \pi \approx 3.14

Answer: π≈3.14\pi \approx 3.14

Worked example 2

Evaluate ∫0π3sin⁡3x dx\int_{0}^{\pi } 3\sin ^{3}x\,dx.
  1. sin⁡3xdx=(1−cos⁡2x)sin⁡xdx\sin ^{3}x dx = (1 - \cos ^{2}x) \sin x dx; u=cos⁡x,du=−sin⁡xdxu = \cos x, du = -\sin x dx
  2. Integral =3∫−111−u2 du= 3\int_{-1}^{1} 1 - u^{2}\,du
  3. 3[43]=43[\dfrac{4}{3}] = 4

Answer: 44

Idea 2 of 3

Even powers and half-angle identities

Use sin⁡2x=1−cos⁡2x2\sin ^{2}x = \frac{1 - \cos 2x}{2} and cos⁡2x=1+cos⁡2x2\cos ^{2}x = \frac{1 + \cos 2x}{2} when both powers are even.

Worked example 1

Evaluate ∫0π22cos⁡2x dx\int_{0}^{\dfrac{\pi }{2}} 2 \cos ^{2}x\,dx.
  1. I0=π2I_{0} = \dfrac{\pi }{2}; In=n−1nInI_{n} = \dfrac{n - 1}{n}I_{n}₋2_{2}
  2. I2=12I0I_2 = \dfrac{1}{2}I_0
  3. 2I2=π2≈1.572I_2 = \dfrac{\pi }{2} \approx 1.57

Answer: π2≈1.57\dfrac{\pi }{2} \approx 1.57

Worked example 2

Evaluate ∫0π23cos⁡4x dx\int_{0}^{\dfrac{\pi }{2}} 3 \cos ^{4}x\,dx.
  1. I0=π2I_{0} = \dfrac{\pi }{2}; In=n−1nInI_{n} = \dfrac{n - 1}{n}I_{n}₋2_{2}
  2. I2=12I0I_2 = \dfrac{1}{2}I_0
  3. I4=34I2I_4 = \dfrac{3}{4}I_2
  4. 3I4=9π16≈1.773I_4 = \dfrac{9\pi }{16} \approx 1.77

Answer: 9π16≈1.77\dfrac{9\pi }{16} \approx 1.77

Idea 3 of 3

Mixed sine and cosine products

Convert a product into a form with a usable derivative or a lower-frequency identity before integrating.

Worked example 1

Evaluate ∫0π2sin⁡2x dx\int_{0}^{\pi } 2\sin ^{2}x\,dx.
  1. sin⁡2x=1−cos⁡2x2\sin ^{2}x = \dfrac{1 - \cos 2x}{2}
  2. F(x)=22(x−sin⁡2x2)F(x) = \dfrac{2}{2}(x - \dfrac{\sin 2x}{2})
  3. F(π)−F(0)=2π2=π≈3.14F(\pi ) - F(0) = \dfrac{2\pi }{2} = \pi \approx 3.14

Answer: π≈3.14\pi \approx 3.14

Worked example 2

Evaluate ∫0π3sin⁡3x dx\int_{0}^{\pi } 3\sin ^{3}x\,dx.
  1. sin⁡3xdx=(1−cos⁡2x)sin⁡xdx\sin ^{3}x dx = (1 - \cos ^{2}x) \sin x dx; u=cos⁡x,du=−sin⁡xdxu = \cos x, du = -\sin x dx
  2. Integral =3∫−111−u2 du= 3\int_{-1}^{1} 1 - u^{2}\,du
  3. 3[43]=43[\dfrac{4}{3}] = 4

Answer: 44

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §8.3,p8.3, p. 524524. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.