First slide

Methods of integration

Question

$\int \frac{x^{5 / 2}}{\sqrt{1 + x^{7}}} d x is$

Easy

A

$\frac{2}{7} \log (x^{7 / 2} + \sqrt{x^{7} + 1}) + c$
B

$\frac{1}{2} \log \frac{x^{7} + 1}{x^{7} - 1} + c$
C

$2 \sqrt{1 + x^{7}} + c$
D

$None of these$

Solution

$Put x^{7 / 2} = t$
$\begin{array}{l} ∴ \frac{7}{2} x^{5 / 2} d x = d t \\ ∴ I = \int \frac{2}{7} \frac{d t}{\sqrt{1 + t^{2}}} = \frac{2}{7} \log (t + \sqrt{1 + t^{2}}) + k \end{array}$
$= \frac{2}{7} \log (x^{7 / 2} + \sqrt{1 + x^{7}}) + k$

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