Can the partial fraction method of integration be used with trig functions contained inside the function to be integrated?

Question

Can the partial fractions method be used to integrate a problem like this? $$\int\frac{1}{\cos(x)\left(\sin^2(x)+4\right)}dx$$ Or, do the trig functions contained inside the denominator ruin it? If yes, would this be the setup: $$\frac{A}{\cos(x)}+\frac{Bx+C}{\sin^2(x)+4}$$

Answer 1

It is better to change a variable first: $$\int\frac{dx}{\cos(x)\left(\sin^2(x)+4\right)}=\int\frac{\cos x \,dx}{\cos^2(x)\left(\sin^2(x)+4\right)}=\left[\begin{array}{c}t=\sin x \\ dt=\cos x\,dx\end{array}\right]=\int\frac{dt}{(1-t^2)\left(t^2+4\right)}$$ Now you can use partial fraction. You can also see that the denominators you'll get are not exactly what you'd expected.