Question
Evaluvate the following intregals:
$\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$
$\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$
✓
Answer
Let $\text{I}=\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$
Let $(5\cos\text{x}+6)=\lambda\frac{\text{d}}{\text{dx}}(2\cos\text{x}+\sin\text{x}+3)+\mu(2\cos\text{x}+\sin\text{x}+3)+\text{v}$
$(5\cos\text{x}+6)=\lambda(-2\sin\text{x}+\cos\text{x})+\mu(2\cos\text{x}+\sin\text{x}+3)+\text{v}$
$(5\cos\text{x}+6)=(-2\lambda+\mu)\sin\text{x}(\lambda+2\mu)\cos\text{x}+(3\mu+\text{v})$
Comparing the coefficient of $\sin\text{x}\ \&\cos\text{x}$ on the both the sides,
$-2\lambda+\mu=0\dots\dots(1)$
$\lambda+2\mu=5\dots\dots(2)$
$3\mu+\text{v}=6\dots\dots(3)$
Solving equations (1), (2) and (3),
$\text{I}=\int\frac{(-2\sin\text{x}+\cos\text{x})}{(2\cos\text{x}+\sin\text{x}+3)}\text{dx}+2\int\text{dx}$
$\text{I}=\log|2\cos\text{x}+\sin\text{x}+3|+2\text{x}+\text{C}$
Let $(5\cos\text{x}+6)=\lambda\frac{\text{d}}{\text{dx}}(2\cos\text{x}+\sin\text{x}+3)+\mu(2\cos\text{x}+\sin\text{x}+3)+\text{v}$
$(5\cos\text{x}+6)=\lambda(-2\sin\text{x}+\cos\text{x})+\mu(2\cos\text{x}+\sin\text{x}+3)+\text{v}$
$(5\cos\text{x}+6)=(-2\lambda+\mu)\sin\text{x}(\lambda+2\mu)\cos\text{x}+(3\mu+\text{v})$
Comparing the coefficient of $\sin\text{x}\ \&\cos\text{x}$ on the both the sides,
$-2\lambda+\mu=0\dots\dots(1)$
$\lambda+2\mu=5\dots\dots(2)$
$3\mu+\text{v}=6\dots\dots(3)$
Solving equations (1), (2) and (3),
$\text{I}=\int\frac{(-2\sin\text{x}+\cos\text{x})}{(2\cos\text{x}+\sin\text{x}+3)}\text{dx}+2\int\text{dx}$
$\text{I}=\log|2\cos\text{x}+\sin\text{x}+3|+2\text{x}+\text{C}$
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