What is the integral of tan(x)2x ?

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

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integral of tan(x)2x

\int tan (x) 2 x d x

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

Solution steps

\int tan (x) \cdot 2 x d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \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})

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

Simplify

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

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

Add a constant to the solution

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

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