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

The integral of x*cos(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 x*cos(npix)

\int x \cdot 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

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

\int_{0}^{1} 6 (x^{2} + 1) e^{- x} d x

\frac{d y}{d t} + 20 y = 24

\int 4 θ sec^{2} (θ) d θ

y^{^{'}} = - \frac{y ^{2}}{( 0.05 + y )}, y (0) = 9

\int \frac{1}{2} x^{\frac{3}{4}} d x

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