What is the integral of xcos(θx) ?

The integral of xcos(θx) is 1/(θ^2)(θxsin(θx)+cos(θx))+C

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integral of xcos(θx)

\int x cos (θ x) d x

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

Solution steps

\int x cos (θ x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{θ ^{2}} d u

= \int \frac{u cos ( u )}{θ ^{2}} d u

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = θ x

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

Simplify

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

Add a constant to the solution

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

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What is the integral of xcos(θx) ?
The integral of xcos(θx) is 1/(θ^2)(θxsin(θx)+cos(θx))+C