Mathematics

# $\int \frac { \cos x + 2 \sin x } { 7 \sin x - 5 \cos x } d x = a x + b \ln | 7 \sin x - 5 \cos x | + c$ then $a+b$ is

$\frac { 13 } { 37 }$

##### SOLUTION
$\displaystyle\int \frac{\cos x+2 \sin x}{7 \sin x- 5 \cos x}$
Let $\displaystyle 2 \sin (x)+ \cos (x)= \frac{9}{74}(7 \sin x- 5 \cos x)+\frac{17}{74}(5 \sin x+ 7 \cos x)$
$\displaystyle \Rightarrow \int \frac{9/74(7 \sin x- 5 \cos x)+17/74(5 \sin x+7 \cos x)}{(7 \sin x- 5 \cos x)}dx$
$\displaystyle \Rightarrow \int \dfrac{9}{74}dx+\frac{17}{74}\int \frac{5 \sin x+7 \cos x}{7 \sin x-5 \cos x}dx$
Let $u= 7 sin x- 5 cos x$
$\therefore dx=\dfrac{1}{5\sin x+7\cos x}du$
$\displaystyle\therefore \frac{9}{74}x+\dfrac{17}{74}\int \dfrac{1}{u}du$
$\dfrac{9}{74}x+\dfrac{17}{74}\log(u)$
$\therefore \dfrac{9}{74}x+\dfrac{17}{74}\log(7 \sin x-5 \cos x)+c$
$\therefore a=\dfrac{9}{74}, b=\dfrac{17}{74}$
$\therefore a+b\Rightarrow \dfrac{26}{74}\Rightarrow \dfrac{13}{37}$

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

#### Realted Questions

Q1 Single Correct Hard
The value of $\displaystyle \int_{0}^{\pi /4}\sqrt{\tan \theta }\: d\theta$ is?
• A. $\displaystyle \frac{\pi }{2\sqrt{2}}\left ( 1+\log \left ( 3-2\sqrt{2} \right ) \right )$
• B. $\displaystyle \frac{\pi }{2\sqrt{2}}\left ( 1+\log \left ( \sqrt{2}-1 \right ) \right )$
• C. $\sqrt{\pi }+\log 2$
• D. None of the above

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate: $\displaystyle\int cos 2\theta.\ln\dfrac{cos \theta +sin\theta}{cos \theta-sin\theta}d\theta$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Solve $\int {\dfrac{1}{{\sqrt[{}]{{9 - 25{x^2}}}}}} dx$
• A. ${{\sin }^{-1}}\left( \dfrac{5x}{3} \right)+C$
• B. $\dfrac{1}{5}{{\sin }^{-1}}\left( \dfrac{3x}{5} \right)+C$
• C. ${{\sin }^{-1}}\left( \dfrac{3x}{5} \right)+C$
• D. $\dfrac{1}{5}{{\sin }^{-1}}\left( \dfrac{5x}{3} \right)+C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium

$\displaystyle \int\frac{\sec x}{\sqrt{\sin(2x+\alpha)+\sin\alpha}}dx$ is equal to
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• C. $\sqrt{2\tan\alpha(\sec x-\sec\alpha)}+c$
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Evaluate $\int { \sin ^{ \tfrac{3}{4} }{ x } } \cos { x } dx$.