Good Evening. I am having a problem with a step in the calculation of the following integral:
$$\int_{0}^x \frac{dt}{\sin^2 t +\cos t} =\int_{0}^x \frac{dt}{-\cos^2 t +\cos t+1} =-\int_{0}^x \frac{dt}{\cos^2 t -\cos t-1} $$
Let $X=\cos t$
$$X^2-X-1=0 \Leftrightarrow X=\frac{1-\sqrt{5}}{2} \mbox{and } X=\frac{1+\sqrt{5}}{2}$$
$$\frac{1}{X^2-X-1}=\frac{\frac{\sqrt{5}}{5}}{X-(\frac{1+\sqrt{5}}{2})}-\frac{\frac{\sqrt{5}}{5}}{X-(\frac{1-\sqrt{5}}{2})}$$
Therefore: $$-\int_{0}^x \frac{dt}{\cos^2 t -\cos t-1}= -\int_{0}^x \frac{\frac{\sqrt{5}}{5}}{\cos t-(\frac{1+\sqrt{5}}{2})}dt + \int_{0}^x \frac{\frac{\sqrt{5}}{5}}{\cos t-(\frac{1-\sqrt{5}}{2})}dt$$
I am having trouble calculating the two last integrals. Please help
Thank you in advance.