What is the integral of xtan(x)) ?

The integral of xtan(x)) is d(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 d (tan (x)) d x

d (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 d tan (x) d x

Take the constant out:

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

= d \cdot \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})

= d (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)

= d (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

= d (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{( 1 - x ^{2} )}{x ^{3}} d x

\int sin (x) e^{1 + c o s (x)} d x

\int 1 x d x

\int \frac{1}{s ^{4} - 2 s ^{3} + s ^{2}} d s

\int \frac{3 x}{6 x ^{2} - 4} d x

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