Evaluate the following: (1+ ) x+ dx - Vidyadip

Question

Evaluate the following:
$\int\frac{(1+\cos\text{x})}{\text{x}+\sin\text{x}}\text{dx}$

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Answer

Consider that, $\text{I}=\int\frac{(1+\cos\text{x})}{\text{x}+\sin\text{x}}\text{dx}$
Let $\text{x}+\sin\text{x}=\text{t}$
$\Rightarrow\ (1+\cos\text{x})\text{dx}=\text{dt}$
Substituting $\text{x}+\sin\text{x}=\text{t}$ and $(1+\cos\text{x})\text{dx}=\text{dt}$ in I, we get
$\text{I}=\int\frac{1}{\text{t}}\text{dt}$
$=\log|\text{t}|+\text{C}$ $\Big[\because\int\frac{1}{\text{x}}\text{dx}=\log|\text{x}|+\text{C}\Big]$
$=\log\big|(\text{x}+\sin\text{x})\big|+\text{C}$ $[\because\ \text{t}=\text{x}+\sin\text{x}]$

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