Evaluate the following integrals: _ 0 ^ 3 3+4 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{3}}\frac{\cos\text{x}}{3+4\sin\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{3}}\frac{\cos\text{x}}{3+4\sin\text{x}}\text{ dx}$
Let $\sin\text{x}=\text{t}$ Then, $\cos\text{x}\text{dx}=\text{dt}$
When $\text{x}=0,\text{t}=0$ and $\text{x}=\frac{\pi}{3},\text{t}=\frac{\sqrt{3}}{2}$
$\therefore\ \text{I}=\int_{0}^\limits{\frac{\pi}{3}}\frac{\cos\text{x}}{3+4\sin\text{x}}\text{ dx}$
$=\int_{0}^\limits{\frac{\sqrt{3}}{2}}\frac{1}{3+4\text{t}}\text{ dt}$
$=\frac{1}{4}\big[\log\big(3+-4\text{t}\big)\big]^{\frac{\sqrt{3}}{2}}_0$
$=\frac{1}{4}\big(\log\big(3+2\sqrt{3}\big)-\log3\big)$
$=\frac{1}{4}\log\Big(\frac{3+2\sqrt{3}}{3}\Big)$

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