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  4. If ∫(( sin x+ sin 3 x+ sin 5 x+ sin 7 x+ sin 9 x+ sin 11 x+ sin 13 x+ sin 15 x/ cos x+ cos 3 x+ cos 5 x+ cos 7 x+ cos 9 x+ cos 11 x+ cos 13 x+ cos 15 x)) d x equals ( ln ( sec mx )/ n ) where m , n ∈ N, find ( m + n ).

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Q. If $\int\left(\frac{\sin x+\sin 3 x+\sin 5 x+\sin 7 x+\sin 9 x+\sin 11 x+\sin 13 x+\sin 15 x}{\cos x+\cos 3 x+\cos 5 x+\cos 7 x+\cos 9 x+\cos 11 x+\cos 13 x+\cos 15 x}\right) d x$ equals
$\frac{\ln (\sec mx )}{ n }$ where $m , n \in N$, find $( m + n )$.

Integrals

Solution:

$\left.I=\int\left[\frac{\sin \left(\frac{8 \cdot 2 x}{2}\right)}{\sin x} \cdot \sin 8 x\right] \div\left(\frac{\sin 8 x}{\sin x} \cdot \cos 8 x\right)\right] d x=\int \tan 8 x d x=\frac{1}{8} \ln (\sec 8 x)+C$