What is the integral of-xcos(nx) ?

The integral of-xcos(nx) is -1/(n^2)(nxsin(nx)+cos(nx))+C

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integral of-xcos(nx)

\int - x cos (n x) d x

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

Solution steps

\int - x cos (n x) d x

Take the constant out:

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

= - \int x cos (n x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{n ^{2}} d u

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = n x

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

Simplify

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

Add a constant to the solution

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

\frac{d}{d x} (cos (3 x))

e^{- s}

5 x - 7

x \to - 3 lim (x^{2} - x - 1)

sec (- 4 x^{2} - 3 x + 2)

What is the integral of-xcos(nx) ?
The integral of-xcos(nx) is -1/(n^2)(nxsin(nx)+cos(nx))+C