Evaluvate the following intregals: 2 +3 3 +4 dx

Question

Evaluvate the following intregals:
$\int\frac{2\sin\text{x}+3\cos\text{x}}{3\sin\text{x}+4\cos\text{x}}\ \text{dx}$

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Answer

Let $\text{I}=\int\frac{2\sin\text{x}+3\cos\text{x}}{3\sin\text{x}+4\cos\text{x}}\ \text{dx}$
Let $(2\sin\text{x}+3\cos\text{x})=\lambda\frac{\text{d}}{\text{dx}}(3\sin\text{x}+4\cos\text{x})+\mu(3\sin\text{x}+4\cos\text{x})+\text{v}$
$(2\sin\text{x}+3\cos\text{x})=\lambda(3\cos\text{x}-4\sin\text{x})+\mu(3\sin\text{x}+4\cos\text{x})+\text{v}$
$(2\sin\text{x}+3\cos\text{x})=(3\lambda+4\mu)\cos\text{x}+(-4\lambda+3\mu)\sin\text{x}+\text{v}$
Compairing the coefficient of $\sin\text{x},\cos\text{x}$ on both the sides,
$3\lambda+4\mu=3\dots\dots(1)$
$-4\lambda+3\mu=2\dots\dots(2)$
$\text{v}=0\dots\dots(3)$
Solving the equation (1), (2) and (3)
$\lambda=\frac{1}{25}$
$\mu=\frac{18}{25}$
$\text{v}=0$
$\text{I}=\frac{1}{25}\int\frac{(3\cos\text{x}-4\sin\text{x})}{(3\sin\text{x}+4\cos\text{x})}\text{dx}+\frac{18}{25}\int\text{dx}$
$\text{I}=\frac{1}{25}\log|3\sin\text{x}+4\cos\text{x}|+\frac{18}{25}\text{x}+\text{C}$

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