Evaluate the Integral integral of xtan(x) with respect to x

Problem

(∫_^)(x*tan(x)*d(x))

Solution

1. Identify the integration method. The integral (∫_^)(x*tan(x)*d(x)) does not have a solution in terms of elementary functions (polynomials, exponentials, logarithms, or standard trigonometric functions).

2. Express the tangent function in terms of complex exponentials using the identity tan(x)=(e(i*x)−e(−i*x))/(i*(e(i*x)+e(−i*x)))

3. Substitute and expand the integrand to prepare for integration using special functions.

4. Integrate term by term. The result involves the polylogarithm function, specifically the dilogarithm (Li_2)(z)

5. Apply the formula for the integral of x*tan(x) which results in a combination of logarithmic and dilogarithm terms.

Final Answer

(∫_^)(x*tan(x)*d(x))=x*ln(1+e(2*i*x))−i/2*x2−i/2*(Li_2)*(−e(2*i*x))+C

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