Why isn't the integrand positive?

Calculus Level 4

$\displaystyle\int_0^1 \sqrt{-\ln x} \quad\mathrm dx=\frac{\pi^{m}}{n}$

If $n$ is an integer and $m$ is a rational number that satisfy the equation above, find $m+n$ .

The answer is 2.5.

1 solution

Siddharth Bhatt
Jun 18, 2015

$y=-\ln x$ $\bbox[8pt, border:1pt solid crimson]{e^y=e^{-\ln x}=e^{\ln\frac{1}{x}}=\frac{1}{x}}$

$\color{#008080}{dx= -\frac{dy}{e^y}}$

$\displaystyle\int_0^1 \sqrt{-\ln x} dx=\displaystyle\int_{\infty}^0 \sqrt{y} \left(-\frac{dy}{e^y}\right)=\displaystyle\int_0^{\infty} e^{-y} y^{\frac{1}{2}}dy=\left(\frac{1}{2}\right)!$

$\bbox[8pt, border:1pt solid crimson]{\left(\frac{1}{2}\right)!=\frac{1}{2} \sqrt{\pi}}$

$\Large{\color{crimson}{\int_0^1 \sqrt{-\ln x} dx=\frac{1}{2} \sqrt{\pi}}}$

Here only $n$ is an integer. Please edit the question accordingly $\ddot\smile$

Karthik Kannan - 5 years, 12 months ago

The question says m,n are integers :/

Samarth Agarwal - 5 years, 12 months ago

I transformed this to gamma function then I did it

Righved K - 5 years, 6 months ago

Why isn't the integrand positive?

The answer is 2.5.

1 solution

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