What is the integral of xcos(yx) ?

The integral of xcos(yx) is 1/(y^2)(yxsin(yx)+cos(yx))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of xcos(yx)

\int x cos (y x) d x

\frac{1}{y ^{2}} (y x sin (y x) + cos (y x)) + C

Solution steps

\int x cos (y x) d x

Apply u-substitution

= \int \frac{u cos ( u )}{y ^{2}} d u

Take the constant out:

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

= \frac{1}{y ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

= \frac{1}{y ^{2}} (u sin (u) - \int sin (u) d u)

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

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

Substitute back

u = y x

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

Simplify

= \frac{1}{y ^{2}} (y x sin (y x) + cos (y x))

Add a constant to the solution

= \frac{1}{y ^{2}} (y x sin (y x) + cos (y x)) + C

What is the integral of xcos(yx) ?
The integral of xcos(yx) is 1/(y^2)(yxsin(yx)+cos(yx))+C