Mathematics

# Evaluate :$\displaystyle\int {\dfrac{9cosx-sinx}{4sinx+5cosx}dx}$

$x+\ln(4\sin x+5\cos x)+c$

##### SOLUTION
$\to \displaystyle \int \dfrac{9\cos x - \sin x}{4\sin x + 5 \cos x} dx$
It is in the form $\dfrac{f(x)}{g(x)}$, so we will write $f(x)$ as $A(g(x)) + B(g'(x))$.
$\to f(x) = 9\cos x - \sin x$
$\to g(x) = 4\sin x + 5 \cos x$
$g'(x) = 4 \cos x - 5 \sin x$
So, $f(x) = g(x) + g'(x)$
$\to \displaystyle \int 1+ \dfrac{g'(x)}{g(x)} dx \Rightarrow x + ln (g(x))+C$
Answer $= x + ln (4\sin x + 5\cos x) + C$

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

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