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

\int \frac{x ^{2} + 7 x - 6}{( x + 1 ) ( x ^{2} - 4 x + 7 )} d x

\int \frac{8}{θ ^{2}} cos (\frac{8}{θ}) d θ

y = ln (8 - 4 x)

f (x) = 21 x

y = 8 x sin (x), (\frac{π}{2}, 4 π)

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