Mathematics

# Evaluate $\displaystyle \int { \dfrac { \sin { \left( x+a \right) } }{ \sin { \left( x+b \right) } } dx }$

##### SOLUTION
$\int {\dfrac{{\sin \left( {x + a} \right)}}{{\sin \left( {x + b} \right)}}} dx$
$putting\,x + b = t$
$dx = dt$
$= \int {\dfrac{{\sin \left( {t - b + a} \right)}}{{\sin t}}} dt$
$= \int {\dfrac{{\sin \left( {t + \left( {a - b} \right)} \right)}}{{\sin t}}} dt$
$= \int {\left[ {\dfrac{{\sin t\cos \left( {a - b} \right)}}{{\sin t}} + \dfrac{{\cos t\sin \left( {a - b} \right)}}{{\sin t}}} \right]} dt$
$= \int {\cos \left( {a - b} \right)dt + \sin \left( {a - b} \right)\int {\cot tdt} }$
$= \cos \left( {a - b} \right)dt + \sin \left( {a - b} \right)\log \sin \left| t \right| + c$
$= \cos \left( {a - b} \right)dt + \sin \left( {a - b} \right)\log \sin \left| {x + b} \right| + c$
$= \left( {x + b} \right)\cos \left( {a - b} \right) + \sin \left( {a - b} \right)\log \sin \left| {x + b} \right| + c$

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 Subjective Medium
Show that $\displaystyle\int \frac{x}{\sqrt{\left ( 4-x^{4} \right )}}dx=\sin ^{-1}\frac{x^{2}}{2}.$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \underset{0}{\overset{\pi}{\int}} \dfrac{x \, dx}{ 1 + \sin \, x}$ =
• A. $\dfrac{\pi}{6}$
• B. $\dfrac{\pi}{3}$
• C. Cannot be valued
• D. $\pi$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Write a value of
$\displaystyle \int { \cfrac { \sin { 2x } }{ { a }^{ 2 }\sin ^{ 2 }{ x } +{ b }^{ 2 }\cos ^{ 2 }{ x } } } dx\quad$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\text { Evaluate: } \int_{0}^{\pi} \dfrac{\mathrm{x} \mathrm{dx}}{\mathrm{a}^{2} \cos ^{2} \mathrm{x}+\mathrm{b}^{2} \sin ^{2} \mathrm{x}}$

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 $\displaystyle f(x)>0 \forall x \in R$