How do you integrate $\int x^{3} cot x d x$ $int x^3 cot x dx$ using integration by parts?

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How do you integrate $\int x^{3} cot x d x$ $int x^3 cot x dx$ using integration by parts?

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Answer 1 · 2016-11-28T18:49:42

You can't.

Explanation:

The antiderivative of $x^{3} cot x$ $x^3cotx$ involves the Polylogarithm Function evaluated at imaginary values.

$I = \int x^{3} cot x d x$ $I = intx^3cotx dx$

Let $u = x^{3}$ $u = x^3$ so $d u = 3 x^{2} d x$ $du = 3x^2 dx$ and

let $d v = cot x d x$ $dv = cotx dx$ so $v = \int cot x d x = ln | sin x |$ $v = intcotx dx = ln abssinx$ .

$I = x^{3} ln | sin x | - 3 \int x^{2} ln | sin x | d x$ $I = x^3lnabssinx - 3int x^2 ln abs sinx dx$

Let $u = x^{2}$ $u = x^2$ so $d u = 2 x d x$ $du = 2x dx$

let $d v = ln | sin x | d x$ $dv = ln abs sinx dx$ so $v = \frac{1}{2} i (x^{2} + L i_{2} (e^{2 i x})) - x ln (1 - e^{2 i x}) + x ln | sin x |$ $v = 1/2i(x^2+Li_2(e^(2ix)))-xln(1-e^(2ix))+xlnabs sinx$

Where $L i_{2} (x) = \infty \sum k = 1 \frac{x^{k}}{k^{2}}$ $Li_2(x) = sum_(k=1)^oo x^k/k^2$ $" "$ (known as the polylogarithm or Jonquiere's function.)

At this point I'll let you finish yourself. (Because I'm out of my depth.)

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How do you integrate $\int x^{3} cot x d x$ $int x^3 cot x dx$ using integration by parts?

1 Answer

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How do you integrate ∫x3cotxdxint x^3 cot x dx using integration by parts?

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How do you integrate $\int x^{3} cot x d x$ $int x^3 cot x dx$ using integration by parts?