How do I integrate $\sin\theta + \sin\theta \tan^2\theta$ ?
First thing,
I have been studying maths for business for approximately 3 months now. Since then, I studied algebra and then I started studying calculus. Yet, my friend stopped me there, and asked me to study Fourier series as we'll need it for our incoming projects. So I feel that I am missing a lot of things as I haven't studied integrals yet.
Today, I encountered this solution that I couldn't understand at all.
Apparently, $\int(\sin\theta + \sin\theta \tan^2\theta)d\theta = \int(\sin\theta (1 + \tan^2\theta))d\theta$.
Below, a link to the solution at 5:08.
https://youtu.be/aw_VM_ZDeIo
He stated the we have to learn the integral identities. So, I started searching the whole internet looking for them. But, I think I couldn't find them. The only thing that I found was something called Magic hexagon. I thought of reading about $\theta$ as it might mean something. But, after all I learned that it is just a regular greek letter used as a variable.
Best Answer
$$ \sin\theta+\sin\theta\tan^2\theta=\sin\theta(1+\tan^2\theta)=\sin\theta\sec^2\theta=\sec\theta\tan\theta $$ because $(1+\tan^2\theta)=\sec^2\theta$ is a 'standard' identity.
Also $$ \frac{d\sec\theta}{d\theta}=\sec\theta\tan\theta $$ is a 'standard' derivative.
Therefore $$ \int(\sin\theta+\sin\theta\tan^2\theta)\, d\theta = \int \sec\theta\tan\theta\, d\theta=\sec\theta+C $$ where $C$ is an arbitrary constant