Evaluvate the following intregals: 4 +5 5 +4 dx - Vidyadip

Question

Evaluvate the following intregals:
$\int\frac{4\sin\text{x}+5\cos\text{x}}{5\sin\text{x}+4\cos\text{x}}\ \text{dx}$

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Answer

Let $\text{I}=\int\frac{4\sin\text{x}+5\cos\text{x}}{5\sin\text{x}+4\cos\text{x}}\ \text{dx}$
$4\sin\text{x}+5\cos\text{x}=\lambda\frac{\text{d}}{\text{dx}}(5\sin\text{x}+4\cos\text{x})+\mu(5\sin\text{x}+4\cos\text{x})+\text{v}$
$4\sin\text{x}+5\cos\text{x}=\lambda(5\cos\text{x}-4\sin\text{x})+\mu(5\sin\text{x}+4\cos\text{x})+\text{v}$
$4\sin\text{x}+5\cos\text{x}=(5\lambda+4\mu)\cos\text{x}+(-4\lambda+5\mu)\sin\text{x}+\text{v}$
Compairing the coefficient of $\sin\text{x}\ \&\cos\text{x}$ on the both the sides,
$-4\lambda+5\mu=4\ ...(1) $
$5\lambda+4\mu=5\ ...(2)$
$\text{v}=0\ ...(3)$
Solving equation (1), (2) and (3),
$\lambda=\frac{9}{41}$
$\mu=\frac{40}{41}$
$\text{v}=0$
$\text{I}=\frac{40}{41}\int\text{dx}+\frac{9}{41}\int\frac{(5\cos\text{x}-4\sin\text{x})}{(5\sin\text{x}+4\cos\text{x})}\ \text{dx}$
$\text{I}=\frac{40}{41}\text{x}+\frac{9}{41}\log|5\sin\text{x}+4\cos\text{x}|+\text{C}$

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