First slide

Evaluation of definite integrals

Question

The value of $\int_{0}^{1} \frac{x^{2 α} - 1}{\log x} d x, if α = \frac{2 n - 1}{2}$ is

Moderate

A

$\log n$
B

$2 \log n$
C

$\log 2 n$
D

$(1 / 2) \log n$

Solution

Let $I = \int_{0}^{1} \frac{x^{2 α} - 1}{\log x} d x$
$\begin{array}{l} {\Rightarrow \frac{d I}{d α} = \int_{0}^{1} \frac{2 x^{2 α} \log x}{\log x} d x = \frac{2}{2 α + 1} x^{2 α + 1}]}_{0}^{1} \\ = \frac{2}{2 α + 1} \\ \Rightarrow I = \log (2 α + 1) + C \end{array}$
When $α = 0, I = 0 \Rightarrow C = 0$
Thus $I = \log (2 α + 1) = \log (2 n)$

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