Evalute the following integrals: 2+3 dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\cos\text{x}}{2+3\sin\text{x}}\text{dx}$

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Answer

Let $\text{I}=\int\frac{\cos\text{x}}{2+3\sin\text{x}}\text{dx}\ .....\text{(i)}$
Let $2+3\sin\text{x}=\text{t}$ then,
$\text{d}(2+3\sin\text{x})=\text{dt}$
$\Rightarrow3\cos\text{x dx}=\text{dt}$
$\Rightarrow\text{dx}=\frac{\text{dt}}{3\cos\text{x}}$
Putting $2+3\sin\text{x}=\text{t and dx}=\frac{\text{dt}}{3\cos\text{x}}$ in equation (i), we get,
$\text{I}=\int\frac{\cos\text{x}}{\text{t}}\times\frac{\text{dt}}{3\cos\text{x}}$
$=\frac{1}{3}\int\frac{\text{dt}}{\text{t}}$
$=\frac{1}{3}\log|\text{t}|+\text{C}$
$=\frac{1}{3}\log|2+3\sin\text{x}|+\text{C}$

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