Question

Evaluate :
$\int \frac{9cosx-sinx}{4sinx+5cosx}dx$

• A. $x+\ln (4\sin x+\cos x)+c$
• B. $x+\ln (\sin x-5\cos x)+c$
• C. $x+\ln (4\sin x+5\cos x)+c$
• D. None of these

Answer 1

The correct option is C $x+\ln (4\sin x+5\cos x)+c$

$\to \int \frac{9\cos x-\sin x}{4\sin x+5\cos x}dx$

It is in the form $\frac{f\left(x\right)}{g\left(x\right)}$, so we will write $f\left(x\right)$ as $A\left(g\right(x\left)\right)+B\left(g^{\prime }\right(x\left)\right)$.

$\to f\left(x\right)=9\cos x-\sin x$

$\to g\left(x\right)=4\sin x+5\cos x$

$g^{\prime }\left(x\right)=4\cos x-5\sin x$

So, $f\left(x\right)=g\left(x\right)+g^{\prime }\left(x\right)$

$\to \int 1+\frac{g^{\prime }\left(x\right)}{g\left(x\right)}dx\Rightarrow x+ln\left(g\right(x\left)\right)+C$

Answer $=x+ln(4\sin x+5\cos x)+C$