What is the integral of (x+4)sin(npix) ?

The integral of (x+4)sin(npix) is -1/(pin)xcos(pinx)+1/(pi^2n^2)sin(pinx)-4/(pin)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+4)sin(npix)

\int (x + 4) sin (nπ x) d x

- \frac{1}{πn} x cos (πn x) + \frac{1}{π ^{2} n ^{2}} sin (πn x) - \frac{4}{πn} cos (πn x) + C

Solution steps

\int (x + 4) sin (nπ x) d x

Expand

(x + 4) sin (nπ x) : x sin (πn x) + 4 sin (πn x)

= \int x sin (πn x) + 4 sin (πn x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x sin (πn x) d x + \int 4 sin (πn x) d x

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

\int 4 sin (πn x) d x = - \frac{4}{πn} cos (πn x)

= - \frac{1}{πn} x cos (πn x) + \frac{1}{π ^{2} n ^{2}} sin (πn x) - \frac{4}{πn} cos (πn x)

Add a constant to the solution

= - \frac{1}{πn} x cos (πn x) + \frac{1}{π ^{2} n ^{2}} sin (πn x) - \frac{4}{πn} cos (πn x) + C

n = 1 \sum \infty 8 (\frac{1}{4})^{n}

\int c e^{c x} d x

\frac{\partial}{\partial x} (r cos (x) sin (y))

\frac{d y}{d t} = k y (1 - \frac{y}{4600})

f (x) = 8 sin (x), g (x) = 8 cos (2 x)

What is the integral of (x+4)sin(npix) ?
The integral of (x+4)sin(npix) is -1/(pin)xcos(pinx)+1/(pi^2n^2)sin(pinx)-4/(pin)cos(pinx)+C