Learning Outcomes
- Compute integrals involving products of trigonometric functions.
Section 5.3 Integration of trigonometry (TI3)
Subsection 5.3.1 Activities
Activity 5.3.1.
Consider \(\displaystyle\int \sin(x)\cos(x) \, dx\text{.}\) Which substitution would you choose to evaluate this integral?
- \(\displaystyle u=\sin(x)\)
- \(\displaystyle u=\cos(x)\)
- \(\displaystyle u=\sin(x)\cos(x)\)
- Substitution is not effective
Activity 5.3.2.
Consider \(\displaystyle\int \sin^4(x)\cos(x) \, dx\text{.}\) Which substitution would you choose to evaluate this integral?
- \(\displaystyle u=\sin(x)\)
- \(\displaystyle u=\sin^4(x)\)
- \(\displaystyle u=\cos(x)\)
- Substitution is not effective
Activity 5.3.3.
Consider \(\displaystyle\int \sin^4(x)\cos^3(x) \, dx\text{.}\) Which substitution would you choose to evaluate this integral?
- \(\displaystyle u=\sin(x)\)
- \(\displaystyle u=\cos^3(x)\)
- \(\displaystyle u=\cos(x)\)
- Substitution is not effective
Activity 5.3.4.
It’s possible to use subtitution to evaluate \(\displaystyle\int \sin^4(x)\cos^3(x) \, dx\text{,}\) by taking advantage of the trigonometric identity \(\sin^2(x)+\cos^2(x)=1\text{.}\)
Complete the following substitution of \(u=\sin(x),du=\cos(x)\,dx\) by filling in the missing \(\unknown\)s.
\begin{align*}
\int \sin^4(x)\cos^3(x)\,dx &=\int\sin^4(x)(\,\unknown\,)\cos(x)\,dx\\
&=\int\sin^4(x)(1-\unknown)\cos(x)\,dx\\
&= \int\unknown(1-\unknown)\,du\\
&= \int (u^4-u^6)\,du\\
&= \frac{1}{5}u^5-\frac{1}{7}u^7+C\\
&= \unknown
\end{align*}
Activity 5.3.5.
Trying to substitute \(u=\cos(x),du=-\sin(x)\,dx\) in the previous example is less successful.
\begin{align*}
\int \sin^4(x)\cos^3(x)\,dx &=-\int\sin^3(x)\cos^3(x)(-\sin(x)\,dx)\\
&=-\int\sin^3(x)u^3\,du\\
&= \cdots?
\end{align*}
Which feature of \(\sin^4(x)\cos^3(x)\) made \(u=\sin(x)\) the better choice?
- The even power of \(\sin^4(x)\)
- The odd power of \(\cos^3(x)\)
Activity 5.3.6.
Try to show
\begin{equation*}
\int \sin^5(x)\cos^2(x)\,dx=
-\frac{1}{7} \, \cos^{7}\left(x\right) + \frac{2}{5} \, \cos^{5}\left(x\right) - \frac{1}{3} \, \cos^{3}\left(x\right)+C
\end{equation*}
by first trying \(u=\sin(x)\text{,}\) and then trying \(u=\cos(x)\) instead.
Which substitution worked better and why?
- \(u=\sin(x)\) due to \(\sin^5(x)\)’s odd power.
- \(u=\sin(x)\) due to \(\cos^2(x)\)’s even power.
- \(u=\cos(x)\) due to \(\sin^5(x)\)’s odd power.
- \(u=\cos(x)\) due to \(\cos^2(x)\)’s even power.
Observation 5.3.7.
When integrating the form \(\int \sin^m(x)\cos^n(x)\,dx\text{:}\)
- If \(\sin\)’s power is odd, rewrite the integral as \(\int g(\cos(x))\sin(x)\,dx\) and use \(u=\cos(x)\text{.}\)
- If \(\cos\)’s power is odd, rewrite the integral as \(\int h(\sin(x))\cos(x)\,dx\) and use \(u=\sin(x)\text{.}\)
Activity 5.3.8.
Let’s consider \(\displaystyle\int \sin^2(x) \, dx\text{.}\)
(a)
Use the fact that \(\sin^2(\theta)=\displaystyle\frac{1-\cos(2\theta)}{2}\) to rewrite the integrand using the above identities as an integral involving \(\cos(2x)\text{.}\)
(b)
Show that the integral evaluates to \(\frac{1}{2} \, x - \frac{1}{4} \, \sin\left(2 \, x\right)+C\text{.}\)
Activity 5.3.9.
Let’s consider \(\displaystyle\int \sin^2(x)\cos^2(x) \, dx\text{.}\)
(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)
Show that integral evaluates to \(\frac{1}{8} \, x - \frac{1}{32} \, \sin\left(4 \, x\right)+C\text{.}\)
Activity 5.3.10.
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 5.3.11.
Consider \(\displaystyle\int \sin^3(x)\cos^5(x) \, dx\text{.}\) Which would be the most useful way to rewrite the integral?
- \(\displaystyle \displaystyle\int (1-\cos^2(x))\cos^5(x) \sin(x)\, dx\)
- \(\displaystyle \displaystyle\int \sin^3(x)\left(\frac{1+\cos(2x)}{2}\right)^2\cos(x) \, dx\)
- \(\displaystyle \displaystyle\int \sin^3(x)(1-\sin^2(x))^2\cos(x) \, dx\)
Remark 5.3.12.
We might also use some other trigonometric identities to manipulate our integrands, listed in Appendix B.
Activity 5.3.13.
Consider \(\displaystyle\int \sin(\theta)\sin(3\theta) \, d\theta\text{.}\)
(a)
Find an identity from Appendix B which could be used to transform our integrand.
(b)
Rewrite the integrand using the selected identity.
(c)
Evaluate the integral.
