First slide

Methods of integration

Question

$If \int \frac{(\sqrt{x})^{5}}{(\sqrt{x})^{7} + x^{6}} d x = a \ln \frac{x^{k}}{x^{k} + 1} + c, the value of a and k, respectively, are$

Moderate

A

$\frac{5}{2} and \frac{2}{5}$
B

$\frac{2}{5} and \frac{5}{2}$
C

$\frac{5}{2} and 2$
D

$None of these$

Solution

$\int \frac{dx}{(\sqrt{x})^{2} + (\sqrt{x})^{7}} = \int \frac{dx}{(\sqrt{x})^{7} (1 + \frac{1}{(\sqrt{x})^{5}})}$
$Put \frac{1}{(\sqrt{x})^{5}} = y, \frac{dy}{dx} = - \frac{5}{2 (\sqrt{x})^{7}}$
$\begin{aligned} ∴ I = \int \frac{- 2 dy}{5 (1 + y)} = - \frac{2}{5} \ln | 1 + y | + c \\ = \frac{2}{5} \ln (\frac{1}{1 + \frac{1}{(\sqrt{x})^{5}}}) ∴ a = \frac{2}{5}, k = \frac{5}{2} \end{aligned}$

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