∫( ( sin 2x/ sin 3x sin 5x) )dx is equal to:

Question

Tardigrade Admin · Accepted Answer

∫( sin 2x/ sin 3x sin 5x)dx=∫( sin (5x-3x)/ sin 3x sin 5x)dx =∫( sin 5x cos 3x- cos 5x sin 3x/ sin 3x sin 5x)dx =∫( cot 3x- cot 5x)dx =(1/3) log | sin 3x|-(1/5) log | sin 5x|+c

Q. $\int (\frac{s i n 2 x}{s i n 3 x s i n 5 x}) d x$ is equal to:

$\frac{1}{5} lo g_{e} ∣ sin 5 x ∣ - \frac{1}{3} lo g_{e} ∣ sin 3 x ∣ + c$

$\frac{1}{3} lo g_{e} ∣ sin 3 x ∣ - \frac{1}{5} lo g_{e} ∣ sin 5 x ∣ + c$

$\frac{1}{3} lo g_{e} ∣ sin 3 x ∣ + \frac{1}{5} lo g_{e} ∣ sin 5 x ∣ + c$

$- \frac{1}{2} cos 2 x + \frac{1}{3} lo g_{e} ∣ sin 3 x ∣$ $+ \frac{1}{5} lo g_{e} ∣ s in 5 x ∣ + c$

$- \frac{1}{2} cos 2 x - \frac{1}{3} lo g_{e} ∣ sin 3 x ∣$ $- \frac{1}{5} lo g_{e} ∣ sin 5 x ∣ + c$

$\int \frac{s i n 2 x}{s i n 3 x s i n 5 x} d x = \int \frac{s i n ( 5 x - 3 x )}{s i n 3 x s i n 5 x} d x$ $= \int \frac{s i n 5 x c o s 3 x - c o s 5 x s i n 3 x}{s i n 3 x s i n 5 x} d x$ $= \int (cot 3 x - cot 5 x) d x$ $= \frac{1}{3} lo g ∣ sin 3 x ∣ - \frac{1}{5} lo g ∣ sin 5 x ∣ + c$

Q. ∫(sin3xsin5xsin2x​)dx is equal to:

51​loge​∣sin5x∣−31​loge​∣sin3x∣+c

31​loge​∣sin3x∣−51​loge​∣sin5x∣+c

31​loge​∣sin3x∣+51​loge​∣sin5x∣+c

−21​cos2x+31​loge​∣sin3x∣ +51​loge​∣sin5x∣+c

−21​cos2x−31​loge​∣sin3x∣ −51​loge​∣sin5x∣+c

∫sin3xsin5xsin2x​dx=∫sin3xsin5xsin(5x−3x)​dx =∫sin3xsin5xsin5xcos3x−cos5xsin3x​dx =∫(cot3x−cot5x)dx =31​log∣sin3x∣−51​log∣sin5x∣+c

Q. $\int (\frac{s i n 2 x}{s i n 3 x s i n 5 x}) d x$ is equal to:

$\frac{1}{5} lo g_{e} ∣ sin 5 x ∣ - \frac{1}{3} lo g_{e} ∣ sin 3 x ∣ + c$

$\frac{1}{3} lo g_{e} ∣ sin 3 x ∣ - \frac{1}{5} lo g_{e} ∣ sin 5 x ∣ + c$

$\frac{1}{3} lo g_{e} ∣ sin 3 x ∣ + \frac{1}{5} lo g_{e} ∣ sin 5 x ∣ + c$

$- \frac{1}{2} cos 2 x + \frac{1}{3} lo g_{e} ∣ sin 3 x ∣$ $+ \frac{1}{5} lo g_{e} ∣ s in 5 x ∣ + c$

$- \frac{1}{2} cos 2 x - \frac{1}{3} lo g_{e} ∣ sin 3 x ∣$ $- \frac{1}{5} lo g_{e} ∣ sin 5 x ∣ + c$

$\int \frac{s i n 2 x}{s i n 3 x s i n 5 x} d x = \int \frac{s i n ( 5 x - 3 x )}{s i n 3 x s i n 5 x} d x$ $= \int \frac{s i n 5 x c o s 3 x - c o s 5 x s i n 3 x}{s i n 3 x s i n 5 x} d x$ $= \int (cot 3 x - cot 5 x) d x$ $= \frac{1}{3} lo g ∣ sin 3 x ∣ - \frac{1}{5} lo g ∣ sin 5 x ∣ + c$