∫((2- sin 2x/1- cos 2x))exdx is equal to

Question

∫((2- sin 2x/1- cos 2x))exdx is equal to (A) - ex cot x+c (B)  ex cot x+c (C)  2ex cot x+c (D)  -2ex cot x+c

Tardigrade Admin · Accepted Answer

Let I=∫((2- sin 2 x/1- cos 2 x)) ex d x =∫((2-2 sin x cos x/2 sin 2 x)) ex d x =∫ undersetII textcosec2 undersetI x ex d x-∫ cot x ex d x =- cot x ex-∫(- cot x) ex d x-∫ cot x ex d x+c =- cot x ex+c

Q. $\int (\frac{2 - s i n 2 x}{1 - c o s 2 x}) e^{x} d x$ is equal to

$- e^{x} cot x + c$

$e^{x} cot x + c$

$2 e^{x} cot x + c$

$- 2 e^{x} cot x + c$

Let $I = \int (\frac{2 - s i n 2 x}{1 - c o s 2 x}) e^{x} d x$
$= \int (\frac{2 - 2 s i n x c o s x}{2 s i n ^{2} x}) e^{x} d x$
$= \int II cosec^{2} I x e^{x} d x - \int cot x e^{x} d x$
$= - cot x e^{x} - \int (- cot x) e^{x} d x - \int cot x e^{x} d x + c$
$= - cot x e^{x} + c$

Q. ∫(1−cos2x2−sin2x​)exdx is equal to

−excotx+c

excotx+c

2excotx+c

−2excotx+c