The integral `int_0^(pi/4)csc x dx` **is improper** because cosecant is not defined on zero (more generally `csc x` is not defined for `x in {k pi, k in ZZ}.`) and the interval of integration includes that point.

The image below shows graph of the function over `[-pi/4,pi/4].` As we can see

`lim_(x to...

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The integral `int_0^(pi/4)csc x dx` **is improper** because cosecant is not defined on zero (more generally `csc x` is not defined for `x in {k pi, k in ZZ}.`) and the interval of integration includes that point.

The image below shows graph of the function over `[-pi/4,pi/4].` As we can see

`lim_(x to 0)csc x=+-infty.` We can also see, from the image, that the above integral will diverge or more precisely it will be equal to infinity.