First slide

Methods of integration

Question

If $\int \frac{(\sqrt{x})^{5} dx}{(\sqrt{x})^{7} + x^{6}} = λlog (\frac{x^{a}}{x^{a} + 1}) + C$ then $a + λ$

Moderate

A

2
B

>2
C

<2
D

>3

Solution

$I = \int \frac{dx}{(\sqrt{x})^{2} + (\sqrt{x})^{7}} = \int \frac{dx}{(\sqrt{x})^{7} (\frac{1}{(\sqrt{x})^{5}} + 1)}$
Put $1 + \frac{1}{(\sqrt{x})^{5}} = t \Rightarrow - \frac{5}{2} \cdot \frac{1}{(\sqrt{x})^{7}} dx = dt$
$\begin{array}{l} I = - \frac{2}{5} \int \frac{dt}{t} = - \frac{2}{5} \log t + C \\ = - \frac{2}{5} \log [1 + \frac{1}{(\sqrt{x})^{5}}] + C = \frac{- 2}{5} \log (\frac{x^{5 / 2} + 1}{x^{5 / 2}}) + C \end{array}$
$\Rightarrow λ = \frac{- 2}{5} and a = \frac{5}{2}$
$∴ a + λ = \frac{21}{10} > 2$

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