First slide

Evaluation of definite integrals

Question

Let $f (x) = \frac{x e^{x}}{(1 + x)^{2}}, x \neq - 1$
Statement-1: The antiderivative $F$ of $f (x)$ satisfying $F (0) = 1$
is $\frac{1}{1 + x} e^{x}$
Statement-2: $f (x)$ is of the form $(g (x) + g^{'} (x)) e^{x}$ for some
$g (x)$ .

Easy

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

$\begin{array}{l} F (x) = \int \frac{x e^{x}}{(1 + x)^{2}} d x \\ = \int (\frac{1}{1 + x} - \frac{1}{(1 + x)^{2}}) e^{x} d x \\ = \frac{1}{1 + x} e^{x} + C \end{array}$
$F (0) = 1 \Rightarrow C = 0$ so $F (x) = \frac{1}{1 + x} e^{x}$ .

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