Correct option is (3) \(4 \log _{e} 3\)
\(I=80 \int\limits_{0}^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16(2 \sin \theta \cdot \cos \theta)}\right) d \theta\)
\(=80 \int\limits_{0}^{\frac{\pi}{4}} \frac{\sin \theta+\cos \theta}{9-16(1-2 \sin \theta \cdot \cos \theta-1)} d \theta\)
\(=80 \int\limits_{0}^{\frac{\pi}{4}} \frac{\sin \theta+\cos \theta}{9+16-16(\sin \theta-\cos \theta)^{2}} d \theta\)
Let \(\sin \theta-\cos \theta=\mathrm{t}\)
\((\cos \theta+\sin \theta) d \theta=d t\)
\(=80 \int\limits_{-1}^{0} \frac{\mathrm{dt}}{25-16 \mathrm{t}^{2}}\)
\(=\frac{80}{16} \int\limits_{-1}^{0} \frac{\mathrm{dt}}{\left(\frac{5}{4}\right)^{2}-\mathrm{t}^{2}}\)
\(\left.=\frac{5}{2\left(\frac{5}{4}\right)} \text{ln} \left|\frac{\frac{5}{4}+t}{\frac{5}{4}-t}\right|\right]_{-1}^{0}\)
\(=2 \text{ln} (1)+4 \text{l}n 3\)
\(=4 \text{ln} 3\)
