IGDTUW 2025 — Mathematics PYQ

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Question

IGDTUW 2025 — Mathematics PYQ

IGDTUW | Mathematics | 2025

Integrate: $\int (\frac{x e ^{x}}{( 1 + x ) ^{2}}) d x$

Choose the correct answer:

Explanation

Solution

To solve this integration, we use the standard calculus identity:

\int e^x [f(x) + f'(x)] dx = e^x f(x) + C

Step 1: Rearrange the Numerator

We can rewrite the $x$ in the numerator as $(1 + x) - 1$ :

\int \frac{((1 + x) - 1)e^x}{(1 + x)^2} dx

Step 2: Split the Fraction

Distribute $e^x$ and split the terms over the common denominator:

\int e^x \left[ \frac{1 + x}{(1 + x)^2} - \frac{1}{(1 + x)^2} \right] dx

\int e^x \left[ \frac{1}{1 + x} - \frac{1}{(1 + x)^2} \right] dx

Step 3: Identify $f(x)$ and $f'(x)$

Let:

$f(x) = \frac{1}{1 + x}$
$f'(x) = -\frac{1}{(1 + x)^2}$

Since the expression is now in the form $\int e^x [f(x) + f'(x)] dx$ , the result is $e^x f(x) + C$ .

\int e^x \left[ \frac{1}{1 + x} + \left( -\frac{1}{(1 + x)^2} \right) \right] dx = \frac{e^x}{1 + x} + C

The correct option is (b) $\left( \frac{e^x}{1 + x} \right) + C$ .