What is the integral of xsin(2nx) ?

The integral of xsin(2nx) is 1/(4n^2)(-2nxcos(2nx)+sin(2nx))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of xsin(2nx)

\int x sin (2 n x) d x

\frac{1}{4 n ^{2}} (- 2 n x cos (2 n x) + sin (2 n x)) + C

Solution steps

\int x sin (2 n x) d x

Apply u-substitution

= \int \frac{u sin ( u )}{4 n ^{2}} d u

Take the constant out:

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

= \frac{1}{4 n ^{2}} \cdot \int u sin (u) d u

Apply Integration By Parts

= \frac{1}{4 n ^{2}} (- u cos (u) - \int - cos (u) d u)

\int - cos (u) d u = - sin (u)

= \frac{1}{4 n ^{2}} (- u cos (u) - (- sin (u)))

Substitute back

u = 2 n x

= \frac{1}{4 n ^{2}} (- 2 n x cos (2 n x) - (- sin (2 n x)))

Simplify

= \frac{1}{4 n ^{2}} (- 2 n x cos (2 n x) + sin (2 n x))

Add a constant to the solution

= \frac{1}{4 n ^{2}} (- 2 n x cos (2 n x) + sin (2 n x)) + C

What is the integral of xsin(2nx) ?
The integral of xsin(2nx) is 1/(4n^2)(-2nxcos(2nx)+sin(2nx))+C