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

The integral of tan(x)(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 tan(x)(x)

\int tan (x) (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 tan (x) x d x

Use the nonelementary integral:

\int tan (x) 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

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