Mathematics

# Evaluate the following integral$\int { \cfrac { \sin { \left( x-\alpha \right) } }{ \sin { \left( x+\alpha \right) } } } dx$

##### SOLUTION
$\displaystyle\int{\dfrac{\sin{\left(x-\alpha\right)}}{\sin{\left(x+\alpha\right)}}dx}$

$=\displaystyle\int{\dfrac{\sin{\left(x+\alpha-2\alpha\right)}}{\sin{\left(x+\alpha\right)}}dx}$

$=\displaystyle\int{\dfrac{\sin{\left(x+\alpha\right)}\cos{2\alpha}-\cos{\left(x+\alpha\right)}\sin{2x}}{\sin{\left(x+\alpha\right)}}dx}$

$=\displaystyle\int{\dfrac{\sin{\left(x+\alpha\right)}\cos{2\alpha}}{\sin{\left(x+\alpha\right)}}dx}-\displaystyle\int{\dfrac{\cos{\left(x+\alpha\right)}\sin{2x}}{\sin{\left(x+\alpha\right)}}dx}$

$=\cos{2\alpha}\displaystyle\int{dx}-\sin{2\alpha}\displaystyle\int{\dfrac{\cos{\left(x+\alpha\right)}}{\sin{\left(x+\alpha\right)}}dx}$

$=x\cos{2\alpha}-\sin{2\alpha}\log{\left|\sin{\left(x+\alpha\right)}\right|}+c$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 One Word Medium
In the equation $\displaystyle \int \frac{dx}{\sqrt{2ax-x^{2}}}= a^{n}\sin ^{-1}\left ( \frac{x}{a}-1 \right ).$
Find the value of n.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Solve
$\int \sqrt{\dfrac{a-x}{x-b}} dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate $\displaystyle\int \dfrac{x^3-1}{x^2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate
$\int\limits_0^\infty {\frac{{\log \left( {1 + {x^2}} \right)dx}}{{1 + {x^2}}}} =$

$\int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx}$