Mathematics

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


ANSWER

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


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 $$cosec^2xdx=-dt$$

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

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

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

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

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

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

Its FREE, you're just one step away


Single Correct Hard Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109
Enroll Now For FREE

Realted Questions

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$None \ of \ these$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
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 $$\displaystyle f(x)>0    \forall  x \in  R$$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives


1 Verified Answer | Published on 17th 08, 2020

View Answer