What is the integral of-xcos(npix) ?

The integral of-xcos(npix) is -1/(pi^2n^2)(pinxsin(pinx)+cos(pinx))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of-xcos(npix)

\int - x cos (nπ x) d x

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

Solution steps

\int - x cos (nπ x) d x

Take the constant out:

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

= - \int x cos (nπ x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = n x π

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

Simplify

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

Add a constant to the solution

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

f (x) = x^{11} e^{x}

\frac{d}{d θ} (2 - sin (θ))

x e^{- x + 1}, at 1

y = \frac{- 2 x}{x ^{2} + 1}, (- 1, 1)

x \to π lim (x^{2} cos (x))

What is the integral of-xcos(npix) ?
The integral of-xcos(npix) is -1/(pi^2n^2)(pinxsin(pinx)+cos(pinx))+C