What is the integral of xcos(yx) ?

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

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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

= \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

: u sin (u) - \int sin (u) d u

= \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

\int \frac{sin ^{2} ( x ) - k}{sin ^{2} ( x )} d x

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\int \frac{x ^{5}}{( x ^{6} + 3 ) ^{\frac{5}{6}}} d x

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