I was trying to evaluate double integral: $\int_{0}^{\pi}dx\int_{x}^{\pi} \frac{\sin y}{y} dy$

I don't know what to do, from double integral calculator the answer is $2$. I checked indefinite integral from it and it is $\operatorname*{Si}(x)+C$. I tried to do it with polar form, but I get nothing interesting.

How to evaluate such integral?

  • 2
    $\begingroup$ Try to change the order of integration (allowed, because the integrand is positive in that area). $\endgroup$ – Professor Vector Jun 11 '17 at 15:38

Switching the order of integration

$$ \int_{0}^{\pi}\mathbb{d}x\int_{x}^{\pi}\mathbb{d}y=\int_{0}^{\pi}\mathbb{d}y\int_{0}^{y}\mathbb{d}x $$

your integral is then

$$ \int_{0}^{\pi}\mathbb{d}y\int_{0}^{y}\frac{\sin(y)}{y}\mathbb{d}x=\int_{0}^{\pi}\sin(y)\mathbb{d}y=2 $$


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