Section 5.3 Integration of trigonometry (TI3)
Learning Outcomes
Compute integrals involving products of trigonometric functions.
Activity 5.3.1.
Consider \(\displaystyle\int \sin(x)\cos(x) \, dx\text{.}\) Which strategy would you choose to evaluate this integral?
The method of substitution
Integration by parts
Other (specify your strategy)
Activity 5.3.2.
Consider \(\displaystyle\int \sin^2(x)\cos(x) \, dx\text{.}\) Which strategy would you choose to evaluate this integral?
The method of substitution
Integration by parts
Other (specify your strategy)
Activity 5.3.3.
Consider \(\displaystyle\int \sin^2(x)\cos^2(x) \, dx\text{.}\) Which strategy would you choose to evaluate this integral?
The method of substitution
Integration by parts
Other (specify your strategy)
Activity 5.3.4.
Let's consider \(\displaystyle\int \sin^2(x)\cos^2(x) \, dx\) again.
(a)
Use the fact that \(\cos^2(\theta)=\displaystyle\frac{1+cos(2\theta)}{2}\) and \(\sin^2(\theta)=\displaystyle\frac{1-cos(2\theta)}{2}\) to rewrite the integrand using the above identities as an integral involving \(\cos^2(2x)\text{.}\)
(b)
Use the above identities to rewrite this new integrand as one involving \(\cos(4x)\text{.}\)
(c)
Evaluate the integral.
Activity 5.3.5.
Consider \(\displaystyle\int \sin^m(x)\cos^n(x) \, dx\text{,}\) where \(m\) is even and \(n\) is odd. Which strategy would you choose to evaluate this integral?
Rewrite sine in terms of cosine by using a double-angle/half-angle identity.
Rewrite cosine in terms of sine by using a double-angle/half-angle identity.
Rewrite cosine in terms of sine by using a pythagorean identity.
Activity 5.3.6.
Consider \(\displaystyle\int \sin^m(x)\cos^n(x) \, dx\text{,}\) where \(m\) and \(n\) are both odd and \(m\geq n\text{.}\) Which strategy would you choose to evaluate this integral?
Rewrite sine in terms of cosine by using a pythagorean identity.
Rewrite cosine in terms of sine by using a double-angle/half-angle identity.
Rewrite cosine in terms of sine by using a pythagorean identity.
Activity 5.3.7.
Consider \(\displaystyle\int \sin^m(x)\cos^n(x) \, dx\text{,}\) where \(m\) and \(n\) are both even and \(m\geq n\text{.}\) Which strategy would you choose to evaluate this integral?
Rewrite sine in terms of cosine by using a pythagorean identity.
Rewrite cosine in terms of sine by using a pythagorean identity.
Rewrite everything in terms of cosine by using a double-angle/half-angle identity.
Activity 5.3.8.
Consider \(\displaystyle\int \sin^4(x)\cos^4(x) \, dx\text{.}\) Which would be the most useful way to rewrite the integral?
\(\displaystyle \displaystyle\int (1-\cos^2(x))^2\cos^4(x) \, dx\)
\(\displaystyle \displaystyle\int \sin^4(x)(1-\sin^2(x))^2 \, dx\)
\(\displaystyle \displaystyle\int \left(\frac{1-\cos(2x)}{2}\right)^2\left(\frac{1+\cos(2x)}{2}\right)^2 \, dx\)
Activity 5.3.9.
Consider \(\displaystyle\int \sin^3(x)\cos^5(x) \, dx\text{.}\) Which would be the most useful way to rewrite the integral?
\(\displaystyle \displaystyle\int (1-\cos^2(x))\sin(x)\cos^5(x) \, dx\)
\(\displaystyle \displaystyle\int \sin^3(x)\left(\frac{1+\cos(2x)}{2}\right)^2\cos(x) \, dx\)
\(\displaystyle \displaystyle\int \sin^3(x)(1-\sin^2(x))^2\cos(x) \, dx\)
Remark 5.3.10.
We might also use some other trigonometric identities to manipulate our integrands, listed in Appendix B.
Activity 5.3.11.
Consider \(\displaystyle\int \sin(\theta)\sin(3\theta) \, d\theta\text{.}\)
Find an identity from Appendix B which could be used to transform our integrand.
Rewrite the integrand using the selected identity.
Evaluate the integral.