What is the integral of x*cos(kx) ?

The integral of x*cos(kx) is 1/(k^2)(kxsin(kx)+cos(kx))+C

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integral of x*cos(kx)

\int x \cdot cos (k x) d x

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

Solution steps

\int x cos (k x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = k x

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

Simplify

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

Add a constant to the solution

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

- 10 x^{4} + x^{3}

y = x^{\frac{13}{14}}

n = 1 \sum \infty ln ((n + 1)^{2})

2 y^{^{''}} + 5 y^{^{'}} - 3 y = 0, y (0) = 1, y^{^{'}} (0) = 4

n = 0 \sum \infty \frac{1}{3 ^{n + 1}}

What is the integral of x*cos(kx) ?
The integral of x*cos(kx) is 1/(k^2)(kxsin(kx)+cos(kx))+C