Evalute the following integrals: 3 +5 dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\sec\text{x}\tan\text{x}}{3\sec\text{x}+5}\text{dx}$

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Answer

Let $\text{I}=\int\frac{\sec\text{x}\tan\text{x}}{3\sec\text{x}+5}\text{dx}$ then,
Putting $\sec\text{x}=\text{t}$
$\Rightarrow\frac{\text{dt}}{\text{dx}}=\sec\text{x}\tan\text{x}$
$\Rightarrow\text{dt}=\sec\text{x}\tan\text{x dx}$
$\therefore\text{I}=\int\frac{\text{dt}}{3\text{t}+5}$
$=\frac{1}{3}\text{ln}|3\text{t}+\text{5}|+\text{C}$
$=\frac{1}{3}\text{ln}|3\sec\text{x}+5|+\text{C}\ \big[\because\text{t}=\sec\text{x}\big]$

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