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How can I change the limit of $\displaystyle\int\limits_{-\pi/2}^{\pi/2} f(x)\,dx$ to $\displaystyle\int\limits_0^{\pi/2} f(x)\,dx$ ?

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    No. Because swapping $x$ with $\pi x/a$ will change the axis of symmetry.2012-10-16

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$\int_o^\pi f(x)dx = \int_0^{\pi/2} 2 f(2y)dy$

by the change of variables $y := x/2$. Is this what you are looking for?