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 ).

Question

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 ). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

.I=∫[( sin ((8 ⋅ 2 x/2))/ sin x) ⋅ sin 8 x] ÷(( sin 8 x/ sin x) ⋅ cos 8 x)] d x=∫ tan 8 x d x=(1/8) ln ( sec 8 x)+C

Q. If $\int (\frac{s i n x + s i n 3 x + s i n 5 x + s i n 7 x + s i n 9 x + s i n 11 x + s i n 13 x + s i n 15 x}{c o s x + c o s 3 x + c o s 5 x + c o s 7 x + c o s 9 x + c o s 11 x + c o s 13 x + c o s 15 x}) d x$ equals
$\frac{l n ( s e c m x )}{n}$ where $m, n \in N$ , find $(m + n)$ .

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

Q. If ∫(cosx+cos3x+cos5x+cos7x+cos9x+cos11x+cos13x+cos15xsinx+sin3x+sin5x+sin7x+sin9x+sin11x+sin13x+sin15x​)dx equals nln(secmx)​ where m,n∈N, find (m+n).

I=∫[sinxsin(28⋅2x​)​⋅sin8x]÷(sinxsin8x​⋅cos8x)]dx=∫tan8xdx=81​ln(sec8x)+C

Q. If $\int (\frac{s i n x + s i n 3 x + s i n 5 x + s i n 7 x + s i n 9 x + s i n 11 x + s i n 13 x + s i n 15 x}{c o s x + c o s 3 x + c o s 5 x + c o s 7 x + c o s 9 x + c o s 11 x + c o s 13 x + c o s 15 x}) d x$ equals
$\frac{l n ( s e c m x )}{n}$ where $m, n \in N$ , find $(m + n)$ .

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