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  4. If displaystyle ∫ (x/x + 1 + ex) d x=px+qln |x + 1 + ex|+c, where c is the constant of integration, then p+q is equal to

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Q. If $\displaystyle \int \frac{x}{x + 1 + e^{x}} d x=px+qln \left|x + 1 + e^{x}\right|+c,$ where $c$ is the constant of integration, then $p+q$ is equal to

NTA AbhyasNTA Abhyas 2022

Solution:

$\displaystyle \int \frac{\left(x + 1 + e^{x}\right) - \left(e^{x} + 1\right)}{x + 1 + e^{x}}dx=\displaystyle \int \left(1 - \frac{\left(e^{x} + 1\right)}{x + 1 + e^{x}}\right)dx=x-ln \left|x + 1 + e^{x}\right|+c$
On comparing,
$p=1,q=-1$
Hence, $p+q=0$