Question

Find $\int {\dfrac{{x{e^x}}}{{{{\left( {1 + x} \right)}^2}}}} dx =$
\eqalign{ & 1)\dfrac{{ - {e^x}}}{{x + 1}} + C \cr & 2)\dfrac{{{e^x}}}{{x + 1}} + C \cr & 3)\dfrac{{x{e^x}}}{{x + 1}} + C \cr & 4)\dfrac{{ - x{e^x}}}{{x + 1}} + C \cr}

Answer 1

Hint : There are functions both in the numerator and in the denominator. Also, they are two different functions, one in exponential and the other one in a quadratic equation form. There is no certain formula to find out the integral of such combined functions. Therefore, we need to solve this question using the Integration by parts method.
The formulas that will be used for solving are:
Integration by parts:
$\int {u.v = u\int {v - \int {\left( {\int v } \right)} } } \left( {\dfrac{{dv}}{{dx}}} \right)$
\eqalign{ & \int {{e^x}} dx = {e^x} + C \cr & \int {\dfrac{1}{x}} dx = \log \left| x \right| + C \cr}

Complete step-by-step answer :
Let, $I = \int {\dfrac{{x{e^x}}}{{{{\left( {1 + x} \right)}^2}}}} dx$
Adding and subtracting by $1$ on the numerator,
$I = \int {\dfrac{{(x + 1 - 1){e^x}}}{{{{\left( {1 + x} \right)}^2}}}} dx$
Grouping the terms on the numerator and denominator, we get
$I = \int {\dfrac{{{e^x}}}{{\left( {1 + x} \right)}}} dx - \int {\dfrac{{{e^x}}}{{{{\left( {1 + x} \right)}^2}}}} dx$
Now, by applying integration by parts,
$I = \dfrac{{{e^x}}}{{\left( {1 + x} \right)}} - \int { - \left( {\dfrac{1}{{{{\left( {1 + x} \right)}^2}}}} \right)} {e^x}dx - \int {\dfrac{{{e^x}}}{{{{(1 + x)}^2}}}} dx + C$
The second term becomes positive,
$I = \dfrac{{{e^x}}}{{\left( {1 + x} \right)}} + \int {\left( {\dfrac{1}{{{{\left( {1 + x} \right)}^2}}}} \right)} {e^x}dx - \int {\dfrac{{{e^x}}}{{{{(1 + x)}^2}}}} dx + C$
The second and the third term are the same, hence they get cancelled.
$I = \dfrac{{{e^x}}}{{\left( {1 + x} \right)}} + C$
Therefore, the final answer is $\dfrac{{{e^x}}}{{\left( {1 + x} \right)}} + C$
Hence, option (2) is the right answer.
So, the correct answer is “Option 2”.

Note : The integration by parts is done in the following order. Inverse trigonometric functions, Logarithmic functions, Algebraic functions, trigonometric functions and lastly the exponential function. It can be remembered as ILATE. Integration by parts consists of both integration and differentiation to be done in the same given problem.