Mathematics

# If the integral $\int \dfrac {5 \tan x}{\tan x-2}dx=x+\alpha In|\sin x-2 \cos x|+k$ , then $a$ is equal to

$2$

##### SOLUTION
$I = \int\dfrac{5 \ tan x}{tan x -2}dx$

$I = \int\dfrac{5 \ \dfrac{sin \ x}{cos\ x}}{\dfrac{sin\ x }{cos\ x} -2}dx\Rightarrow \int\dfrac{5 sin x}{sinx-2cos x}dx$

$\begin{bmatrix}\because\displaystyle 5\ sin\ x\leftrightarrow sin\ x - 2\ cos x +2\ cos \ x+ 4\ sin\ x\end{bmatrix}$

$I = \int\begin{pmatrix}\dfrac{sin\ x - 2\ cos\ x}{sin\ x-2\ cos \ x}+\dfrac{4\ sin\ x+2\ cos\ x}{sin\ x-2\ cos\ x}\end{pmatrix}dx$

$I = \int\begin{pmatrix}\dfrac{sin\ x - 2\ cos\ x}{sin\ x-2\ cos \ x}\end{pmatrix}dx+\int\begin{pmatrix}\dfrac{4\ sin\ x+2\ cos\ x}{sin\ x-2\ cos\ x}\end{pmatrix}dx$

$I=I_1+I_2$ ..............(i)

$I_1 = \int dx \Rightarrow x+c_1$

$I_2=\int\dfrac{2\ (2\ sin \ x+cos\ x)}{sin\ x-2\ cos\ x}dx$

put $sin\ x-2\ cos\ x=t\rightarrow(2\ sin\ x+cos\ x)dx= dt$

$\Rightarrow \int\dfrac{2}{t}dt$

$\Rightarrow2\ ln\begin{vmatrix}t\end{vmatrix}+c_2$
from (i) we get,

$I=x+ 2\ ln\begin{vmatrix}sin\ x-2\ cos\ x\end{vmatrix}+ C$........(ii)

$I=x+ \alpha\ ln\begin{vmatrix}sin\ x-2\ cos\ x\end{vmatrix}+ C$...........(iii)

from (ii) and (iii)

$\alpha= 2$

$\therefore \text{correct option is D}$

Its FREE, you're just one step away

Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle \int x\sin x\sec ^{3}x dx$ is equal to
• A. $\displaystyle \frac{1}{2}\left [ \sec^{2}x-\tan x \right ]+c$
• B. $\displaystyle \frac{1}{2}\left [ x\sec^{2}x+\tan x \right ]+c$
• C. $\displaystyle \frac{1}{2}\left [\sec^{2}x+\tan x \right ]+c$
• D. $\displaystyle \frac{1}{2}\left [ x\sec^{2}x-\tan x \right ]+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\displaystyle \int { { e }^{ 3\log { x } }{ \left( { x }^{ 4 }+1 \right) }^{ -1 } } dx=$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\displaystyle\int^4_0\dfrac{dx}{\sqrt{x^2+2x+3}}$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate:
$\int { \left[ \sin { \left( \log { x } \right) } +\cos { \left( \log { x } \right) } \right] } dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$4 \displaystyle \int \dfrac{\sqrt{a^6 + x^8}}{x} dx$ is equal to ______________________.
• A. $a^6 ln\vert \dfrac {\sqrt{a^6 + x^8} - a^3} {\sqrt {a^6 + x^8} + a^3}\vert + c$
• B. $\sqrt{a^6 + x^8} + \dfrac {a^3}{2} ln \vert \dfrac {\sqrt{a^6 + x^8} - a^3} {\sqrt {a^6 + x^8} + a^3}\vert + c$
• C. $a^6 ln \vert \dfrac {\sqrt{a^6 + x^8} + a^3} {\sqrt {a^6 + x^8} - a^3}\vert + c$
• D. $\sqrt{a^6 + x^8} + \dfrac{a^3}{2} ln \vert \dfrac {\sqrt{a^6 + x^8} + a^3} {\sqrt {a^6 + x^8} - a^3}\vert + c$