Q35 / 116 · NTA Abhyas (JEE Main) 2020 · Mathematics · Definite Integration

The integral $I = \int_{e}^{e + 1} \frac{1 + x^{2}}{1 + x^{3}} d x$ satisfies

A
$I > 2$
B
$I > e$
C
$I < 0$
Correct answer
D
$I < 1$

Solution

x \in (e, e + 1)

\Rightarrow x^{2} < x^{3}

\Rightarrow \frac{1 + x^{2}}{1 + x^{3}} < 1

\Rightarrow \int_{e}^{e + 1} \frac{1 + x^{2}}{1 + x^{3}} d x < \int_{e}^{e + 1} 1 d x

\Rightarrow I < {[x]}_{e}^{e + 1}

\Rightarrow I < 1

Hence, the correct answer is option: 'D'.

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