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


ANSWER

$$\frac { 13 } { 37 }$$


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
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