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Exercise:
Solve the following egral: _^infty fracx^x^+x^+ddx H: x^+x^+x^-x+x^+x+

Solution:
Since the egral is even we can rewrite it as _^infty fracx^x^+x^+ddx frac_-infty^infty fracx^x^+x^+ddx. Observe that x^+x^+ does not have any real roots and that textdegx^+x^+geq textdegx^ so we can use a stated result _-infty^infty fracx^x^+x^+ddx pi i_textIm textResfz Using the h the poles of f are given by: z_frac+sqrti z_frac-sqrti z_frac-+sqrti z_frac--sqrti From the formula we know that only z_ and z_ are relevant for the calculation so we find those residues: textResfz_ lim_zrightarrow z_fracz^leftz-frac-sqrtirightleftz-frac-+sqrtirightleftz-frac--sqrtiright fracfrac+sqrti-sqrti +sqrti fracfrac+sqrti-sqrti- frac-+sqrtisqrti- fracsqrt i + sqrt i + frac-sqrti--+sqrti- - frac-+sqrti- frac-sqrti frac-fracsqrti Analogously we get for z_: textResfz_ -frac-fracsqrti So we get: _^infty fracx^x^+x^+ddx frac_-infty^infty fracx^x^+x^+ddx pi ifracleftfrac-fracisqrt-frac-fracisqrtright fracpisqrt
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Exercise:
Solve the following egral: _^infty fracx^x^+x^+ddx H: x^+x^+x^-x+x^+x+

Solution:
Since the egral is even we can rewrite it as _^infty fracx^x^+x^+ddx frac_-infty^infty fracx^x^+x^+ddx. Observe that x^+x^+ does not have any real roots and that textdegx^+x^+geq textdegx^ so we can use a stated result _-infty^infty fracx^x^+x^+ddx pi i_textIm textResfz Using the h the poles of f are given by: z_frac+sqrti z_frac-sqrti z_frac-+sqrti z_frac--sqrti From the formula we know that only z_ and z_ are relevant for the calculation so we find those residues: textResfz_ lim_zrightarrow z_fracz^leftz-frac-sqrtirightleftz-frac-+sqrtirightleftz-frac--sqrtiright fracfrac+sqrti-sqrti +sqrti fracfrac+sqrti-sqrti- frac-+sqrtisqrti- fracsqrt i + sqrt i + frac-sqrti--+sqrti- - frac-+sqrti- frac-sqrti frac-fracsqrti Analogously we get for z_: textResfz_ -frac-fracsqrti So we get: _^infty fracx^x^+x^+ddx frac_-infty^infty fracx^x^+x^+ddx pi ifracleftfrac-fracisqrt-frac-fracisqrtright fracpisqrt
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ca, complex, cos, integral, substitution
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ENG (English)
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