Does Wolfram Alpha allow the calculation of line integrals, and if so, what would be the format to enter it in?
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Calculator for Line Integrals
Does Wolfram Alpha have a line integral calculator?
Is it possible in Wolfram Alpha to calculate a line integral, and if so, what is the form that has to be used? Wolfram Alpha line integrals
In this case, MathPhD is right (as we would hope with a handle such as this): this integral is a very neat application of complex variables to real-valued functions. In a first course on complex analysis, It is a standard application, but it is really quite fascinating.  describes this process in more detail than I care to, but in a nutshell, the idea goes like this: * Treat the integrand as a function of a complex variable, and note that, on the real axis, cos z/[(z^2 + 1)(z^2 + 2)] = Re e^(iz)/[(z^2 + 1)(z^2 + 2)] = Re f(z) * Consider the contour integral of f(z) around a semicircle, centered at the origin and oriented counter-clockwise, of radius R (if you’ve studied any multivariable calculus, this is analogous to a path or line integral in two real dimensions). * Use your complex variable theorem of choice (a calculation of residues  together with the Cauchy residue theorem  will simplify the contour integral of f around the semicircle C_R). In this integral there are two parts: a semicircular arc integral with nonzero imaginary part traced between (0, pi) and (R, 0) parameterized by (R cos t, R sin t), and an integral over a real line segment. In the context you’re accustomed to, the integral over a real segment is the same as the real integral. Infinity is the limit of this integral. We can show that because e*(iz) is bounded by 1, the contour integral over the circular part goes to 0 as R –> infinity. As a result, the complex contour integral derives its value exclusively from the integral over the real axis. Taking the real part of the integral on that axis yields the value of the desired integral, since the integral is nonzero on that axis.