Mathematics

# The value of $\int {\dfrac{{dx}}{{\sin x.\sin \left( {x + \alpha } \right)}}}$ is equal to

##### SOLUTION
$\displaystyle I=\int \dfrac{dx}{ \sin x . \sin (x+\alpha)}$

$\displaystyle =\dfrac{1}{\sin a} \int \dfrac{\sin (x+a-x)}{\sin x. \sin (x+a)} dx$

$\displaystyle =\dfrac{1}{ \sin a} \int \dfrac{ \sin (x+a) \cos x- \sin x\cos (x+a)}{\sin x . \sin (x+a)}dx$

$\displaystyle =\dfrac{1}{\sin a} \int [\cot x dx- \cot (x+a)] dx$

$=\dfrac{1}{\sin a} [\log |\sin x|-\log |\sin (x+a)| ] +c$

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

#### Realted Questions

Q1 Single Correct Medium
Let $I={\int}_{0}^{1}\dfrac{\sin x}{\sqrt{x}}dx$ and $J={\int}_{0}^{1}\dfrac{\cos x}{\sqrt{x}}dx$. Then, which one of the following is true?
• A. $I<\dfrac{2}{3}$ and $J<2$
• B. $I<\dfrac{2}{3}$ and $J>2$
• C. $I>\dfrac{2}{3}$ and $J<2$
• D. $I>\dfrac{2}{3}$ and $J>2$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int { { e }^{ x } } \left[ \tan { x } -\log { \left( \cos { x } \right) } \right] dx=$
• A. ${ e }^{ x }\log { \left( \ cosec { x } \right) } +c$
• B. ${ e }^{ x }\log { \left( \cos { x } \right) } +c$
• C. ${ e }^{ x }\log { \left( \sin { x } \right) } +c$
• D. ${ e }^{ x }\log { \left( \sec { x } \right) } +c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int { \sin ^{ 3 }{ \left( 2x+1 \right) } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\int {{e^x}\left( {{{\sec }^2}x + \tan x} \right)} dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard

In calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts.

$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$ $\int\, u^{n}(x)v_{n}(x)\, dx$ where $v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$

Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration  by parts is especially useful when calculating $\int P_{n}(x)\, Q(x)\, dx$, where $P_{n}(x)$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.