Mathematics

# $\displaystyle \int \dfrac {dx}{\sin x. \sin (x + a)}$ is equal to

$cosec \, a \, ln \left |\dfrac {\sin x}{\sin (x + a)}\right | + C$

##### SOLUTION
$\displaystyle\int \dfrac{dx}{\sin x\cdot \sin (x+a)}$

$=\displaystyle\int \dfrac{dx}{\sin x(\sin x\cos a+\cos x\sin a)}$

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

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

$=\displaystyle\int \dfrac{cosec^2xdx}{(\cos a+\cot x\sin a)}$

Let $\cot x=t$ then $cosec^2xdx=-dt$

$=-\displaystyle\int \dfrac{dt}{\cos a+\sin at}$  ...........   $\left(\because \displaystyle\int \dfrac{1}{x}dx=ln x+c\right)$

$=-\dfrac{1}{\sin a}log|\cos a+\sin at|+c$

$=-cosec\ a\ log|\cos a+\sin a\cot x|+c$

$=-cosec\ a\ log\left|\cos a+\dfrac{\sin a\times \cos x}{\sin x}\right|+c$

$=-cosec\ a\ log\left|\dfrac{\sin x\cos a+\sin a\cos x}{\sin x}\right|+c$

$=+cosec\ a\ log\left|\dfrac{\sin x}{\sin (x+a)}\right|+c$    ...................$\left(\because -log x=log \dfrac{1}{x}\right)$.

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Subjective Medium
Find the anti-derivative of $(ax+b)^2$ with respect to $x$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Obtain: $\int { \cfrac { (3x+2) }{ \left( x+1 \right) \left( x+2 \right) \left( x-3 \right) } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following definite integrals :
$\displaystyle \int _{0}^{\pi} \dfrac {1}{1+\sin x}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Solve
$\int {\dfrac {x-2}{x ^2 - 4x + 3}} dx =$
• A. $log \sqrt {x ^2 - 4x + 3 + c}$
• B. $xlog (x - 3) - 2 log (x - 2) + c$
• C. $log [(x - 3)(x - 1)]$
• D. $None \ of \ these$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$