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

= \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

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

= \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

\frac{\partial}{\partial x} (z e^{x yz})

\frac{1}{1 - x}

x \cdot \frac{d y}{d x} = y^{2} - y

x \to 2 lim (4 x^{2} + x - 4)

f (t) = cos (4 t)

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