What is the integral of x*tan(x) ?

The integral of x*tan(x) is i 1/2 \Li_{2}(-e^{2ix})+i 1/2 x^2-xln(1+cos(2x)+isin(2x))+C

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integral of x*tan(x)

\int x \cdot tan (x) d x

i \frac{1}{2} Li_{2} (- e^{2 i x}) + i \frac{1}{2} x^{2} - x ln (1 + cos (2 x) + i sin (2 x)) + C

Solution steps

\int x tan (x) d x

Use the nonelementary integral:

\int x tan (x) d x = i \frac{1}{2} Li_{2} (- e^{2 i x}) + i \frac{1}{2} x^{2} - x ln (1 + e^{2 i x})

= i \frac{1}{2} Li_{2} (- e^{2 i x}) + i \frac{1}{2} x^{2} - x ln (1 + e^{2 i x})

Apply imaginary number rule:

e^{ia} = cos (a) + i sin (a)

= i \frac{1}{2} Li_{2} (- e^{2 i x}) + i \frac{1}{2} x^{2} - x ln (1 + cos (2 x) + i sin (2 x))

Add a constant to the solution

= i \frac{1}{2} Li_{2} (- e^{2 i x}) + i \frac{1}{2} x^{2} - x ln (1 + cos (2 x) + i sin (2 x)) + C

\frac{\partial}{\partial x} (\frac{x}{1 + x})

3 (x)^{2} + 8 (x) + 12

x \to 2 lim (\frac{x}{x ^{2} + 1})

y^{^{''}} + 4 y^{^{'}} - 2 y = 2 x^{2} - 3 x + 6

\int \frac{- 3}{( 2 x ) ^{2}} d x

What is the integral of x*tan(x) ?
The integral of x*tan(x) is i 1/2 \Li_{2}(-e^{2ix})+i 1/2 x^2-xln(1+cos(2x)+isin(2x))+C