- #1
Frogeyedpeas
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This thing is killing me! Integration Help
So I was doing some research and came across the following indefinite integral:
∫tan(x)ln(x) dx
where the domain of x is the complex plane so this can be re-written as:
∫tan(z)ln(z) dz...
So I began solving the problem like so:
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln(cos(x))/x dx (integration by parts)
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + e^-ix)/2)/x dx (exponential definition of cos)
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + e^-ix))/x dx - ln(2)ln(x) + C (properties of logarithms)
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix)^2 + 1)/x dx - ix - ln(2)ln(x) + C (property of logarithms along with combining terms in the initial fraction)
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + i)(e^ix - i))/x dx - ix - ln(2)ln(x) + C (expanding the sum of squares)
Which leaves us with this as the remaining problem:
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + i))/x dx + ∫ln((e^ix - i))/x dx - ix - ln(2)ln(x) + C
So how on Earth do you solve these two problems:
∫ln((e^ix + i)/x dx
∫ln(e^ix - i))/x dx
I tried using Wolfram Mathematica but it could not integrate this problem and I can't imagine how a discrete method would work on this.
So I was doing some research and came across the following indefinite integral:
∫tan(x)ln(x) dx
where the domain of x is the complex plane so this can be re-written as:
∫tan(z)ln(z) dz...
So I began solving the problem like so:
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln(cos(x))/x dx (integration by parts)
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + e^-ix)/2)/x dx (exponential definition of cos)
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + e^-ix))/x dx - ln(2)ln(x) + C (properties of logarithms)
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix)^2 + 1)/x dx - ix - ln(2)ln(x) + C (property of logarithms along with combining terms in the initial fraction)
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + i)(e^ix - i))/x dx - ix - ln(2)ln(x) + C (expanding the sum of squares)
Which leaves us with this as the remaining problem:
∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + i))/x dx + ∫ln((e^ix - i))/x dx - ix - ln(2)ln(x) + C
So how on Earth do you solve these two problems:
∫ln((e^ix + i)/x dx
∫ln(e^ix - i))/x dx
I tried using Wolfram Mathematica but it could not integrate this problem and I can't imagine how a discrete method would work on this.