Evaluvate the following intregals 1 4+3 dx - Vidyadip

Question

Evaluvate the following intregals
$\int\frac{1}{4+3\tan\text{x}}\ \text{dx}$

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Answer

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

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