What is the integral from y to 2 of sin((pix)/(2y)) ?

The integral from y to 2 of sin((pix)/(2y)) is -(2ycos(pi/y))/pi

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from y to 2 of sin((pix)/(2y))

\int_{y}^{2} sin (\frac{π x}{2 y}) d x

- \frac{2 y cos ( \frac{π}{y} )}{π}

Solution steps

\int_{y}^{2} sin (\frac{π x}{2 y}) d x

Apply u-substitution

: \int_{\frac{π}{2}}^{\frac{π}{y}} \frac{2 y sin ( u )}{π} d u

= \int_{\frac{π}{2}}^{\frac{π}{y}} \frac{2 y sin ( u )}{π} d u

Take the constant out:

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

= \frac{2 y}{π} \cdot \int_{\frac{π}{2}}^{\frac{π}{y}} sin (u) d u

Use the common integral:

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

= \frac{2 y}{π} [- cos (u)]_{\frac{π}{2}}^{\frac{π}{y}}

Compute the boundaries:

- cos (\frac{π}{y})

= \frac{2 y}{π} (- cos (\frac{π}{y}))

Simplify

= - \frac{2 y cos ( \frac{π}{y} )}{π}

\int_{1}^{2} 3 x^{5} d x

\int_{1}^{3} (arctan (x)) d x

\int_{- \infty}^{2} (x - 2) e^{x} d x

\int_{- 1}^{1} e^{x} - x^{2} + 2 d x

\int_{- 1}^{1} cos (x^{2}) d x

What is the integral from y to 2 of sin((pix)/(2y)) ?
The integral from y to 2 of sin((pix)/(2y)) is -(2ycos(pi/y))/pi