(a) Find: ∫ (cos x)/((4+ sin^2 x)(5-4 cos^2 x)) d x

\[\begin{aligned} I & =\int \frac{\cos x}{\left(4+\sin ^{2} x\right)\left(5-4 \cos ^{2} x\right)} d x \\ & =\int \frac{\cos x}{\left(4+\sin ^{2} x\right)\left(1+4 \sin ^{2} x\right)} d x \end{aligned} \] \[\begin{aligned} I & =\int \frac{d t}{\left(4+t^{2}\right)\left(1+4 t^{2}\right)} \\ & =-\frac{1}{15} \int \frac{d t}{4+t^{2}}+\frac{4}{15} \int \frac{d t}{1+4 t^{2}} \quad(\because \text { using Partial Fraction }) \\ & =-\frac{1}{30} \tan ^{-1}\left(\frac{t}{2}\right)+\frac{2}{15} \tan ^{-1}(2 t)+C \\ & =-\frac{1}{30} \tan ^{-1}\left(\frac{\sin x}{2}\right)+\frac{2}{15} \tan ^{-1}(2 \sin x)+C \end{aligned} \] \[\begin{aligned} & I=\int_{0}^{\pi} \frac{d x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x} \\ &=2 \int_{0}^{\pi / 2} \frac{d x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x} \\ &=2 \int_{0}^{\pi / 2} \frac{\sec ^{2} x}{a^{2}+b^{2} \tan ^{2} x} d x \\ & \tan x=t \text { gives } \\ & I=2 \int_{0}^{\infty} \frac{d t}{a^{2}+b^{2} t^{2}} \\ &\left.=\frac{2}{b^{2}} \cdot \frac{b}{a} \tan ^{-1}\left(\frac{b t}{a}\right)\right]_{0}^{\infty} \\ &=\frac{\pi}{a b} \end{aligned} \]