Integrating Trig Functions

[dagr8est]

Sorry, this is kind of hard to type in computer language so bare with me. Pretend that "S" is the integral symbol.

Question:
Solve the indefinite integral: S sin^2(x/4)dx

Let x/4 = a
da/dx = 1/4
da = 1/4dx

S sin^2(x/4)dx
= 4 S sin^2(a)da

Is that the correct substitution and what am I supposed to do after that?

 

[soroban]

Hello, dagr8est!


With \(\displaystyle \sin^2\theta\) or \(\displaystyle \cos^2\theta\), we need these identities:

\(\displaystyle \L\;\;\;[1]\;\sin^2\theta \:=\:\frac{1\,-\,\cos2\theta}{2}\;\;\;\;\;[2]\;\cos^2\theta\:=\:\frac{1\,+\,2\cos\theta}{2}\)

\(\displaystyle \L\int \sin^2\left(\frac{x}{4}\right)\,dx\)

Using [1], we have: \(\displaystyle \L\:\frac{1}{2}\int\left[1\,-\,\cos\left(\frac{x}{2}\right)\right]\,dx \;=\;\frac{1}{2}\int dx \,-\,\frac{1}{2}\int\cos\left(\frac{x}{2}\right)\,dx\)

Then for the second integral, let \(\displaystyle \L\, u \,=\,\frac{x}{2}\)