What is the integral of xtan(x) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of xtan(x)

\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 + 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

\int_{1}^{e} 5 x^{2} ln (x) d x

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

x \to 0 lim (x^{n} sin (\frac{1}{x}))

\int 1 + \frac{1}{4} x d x

(2.4) (3.6)

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