Hi,
I am having trouble at transforming trigonometric integrands. I just need these to be transformed then I think I will be able to integrate it. I hope you can help me.
Thank you.
$\displaystyle \displaystyle \begin{align*} \int{\frac{3\cos{x} - 4}{\sin^2{x}}\,dx} &= \int{\frac{3\cos{x}}{\sin^2{x}}\,dx} - \int{\frac{4}{\sin^2{x}}\,dx} \\ &= \int{\frac{3\cos{x}}{\sin^2{x}}\,dx} - \int{4\csc^2{x}\,dx}\end{align*} $
Solve the first using the substitution $\displaystyle \displaystyle \begin{align*} u = \sin{x} \implies du = \cos{x}\,dx \end{align*} $ and for the second, what is the derivative of $\displaystyle \displaystyle \begin{align*} \cot{x} \end{align*} $?
$\displaystyle \displaystyle \begin{align*} \int{\left(3 - \csc^2{y}\right)\cos{y}\,dy} &= \int{\left(3 - \frac{1}{\sin^2{y}}\right)\cos{y}\,dy} \end{align*} $
Now make the substitution $\displaystyle \displaystyle \begin{align*} u = \sin{y} \implies du = \cos{y}\,dy \end{align*} $.