NIMCET 2014 — Mathematics PYQ

Solution

Concept:

• Integration by substitution: If $x=f(t)$, then $dx=f^{\prime }(t)dt$.

Calculation:

Let $x{e}^x+1=t\Rightarrow (x{e}^x+e^x)dx=dt\Rightarrow e^x(x+1)dx=dt$

$I=\int \frac{1}{(x{e}^x)t}dt=\int \frac{1}{(t-1)t}dt$

$I=\int \left(\frac{1}{t-1}-\frac{1}{t}\right)dt=\log (t-1)-\log t+C$

$I=\log \left(\frac{t-1}{t}\right)+C$

Substituting back $t=x{e}^x+1$:

$I=\log \left(\frac{x{e}^x}{1+x{e}^x}\right)+C$

Correct Option: 4