What is the integral of xcos(ax) ?

The integral of xcos(ax) is 1/(a^2)(axsin(ax)+cos(ax))+C

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integral of xcos(ax)

\int x cos (a x) d x

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

Solution steps

\int x cos (a x) d x

Apply u-substitution

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

Substitute back

u = a x

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

Simplify

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

Add a constant to the solution

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

What is the integral of xcos(ax) ?
The integral of xcos(ax) is 1/(a^2)(axsin(ax)+cos(ax))+C