What is the integral from 0 to 1/2 of xcos((pikx)/2) ?

The integral from 0 to 1/2 of xcos((pikx)/2) is (piksin((pik)/4)+4cos((pik)/4)-4)/(pi^2k^2)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 1/2 of xcos((pikx)/2)

\int_{0}^{\frac{1}{2}} x cos (\frac{πk x}{2}) d x

\frac{πk sin ( \frac{πk}{4} ) + 4 cos ( \frac{πk}{4} ) - 4}{π ^{2} k ^{2}}

Solution steps

\int_{0}^{\frac{1}{2}} x cos (\frac{πk x}{2}) d x

Apply u-substitution

: \int_{0}^{\frac{πk}{2}} \frac{u cos ( \frac{u}{2} )}{π ^{2} k ^{2}} d u

= \int_{0}^{\frac{πk}{2}} \frac{u cos ( \frac{u}{2} )}{π ^{2} k ^{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} k ^{2}} \cdot \int_{0}^{\frac{πk}{2}} u cos (\frac{u}{2}) d u

Apply u-substitution

: \int_{0}^{\frac{πk}{4}} 4 v cos (v) d v

= \frac{1}{π ^{2} k ^{2}} \cdot \int_{0}^{\frac{πk}{4}} 4 v cos (v) d v

Take the constant out:

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

= \frac{1}{π ^{2} k ^{2}} \cdot 4 \cdot \int_{0}^{\frac{πk}{4}} v cos (v) d v

Apply Integration By Parts

: [v sin (v) - \int sin (v) d v]_{0}^{\frac{πk}{4}}

= \frac{1}{π ^{2} k ^{2}} \cdot 4 [v sin (v) - \int sin (v) d v]_{0}^{\frac{πk}{4}}

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

= \frac{1}{π ^{2} k ^{2}} \cdot 4 [v sin (v) - (- cos (v))]_{0}^{\frac{πk}{4}}

Simplify

\frac{1}{π ^{2} k ^{2}} \cdot 4 [v sin (v) - (- cos (v))]_{0}^{\frac{πk}{4}} : \frac{4}{π ^{2} k ^{2}} [v sin (v) + cos (v)]_{0}^{\frac{πk}{4}}

= \frac{4}{π ^{2} k ^{2}} [v sin (v) + cos (v)]_{0}^{\frac{πk}{4}}

Compute the boundaries:

\frac{πk}{4} sin (\frac{πk}{4}) + cos (\frac{πk}{4}) - 1

= \frac{4}{π ^{2} k ^{2}} (\frac{πk}{4} sin (\frac{πk}{4}) + cos (\frac{πk}{4}) - 1)

Simplify

= \frac{πk sin ( \frac{πk}{4} ) + 4 cos ( \frac{πk}{4} ) - 4}{π ^{2} k ^{2}}

\int_{0}^{3} \frac{e ^{x}}{4 + e ^{x}} d x

\int_{0}^{3} 3 y - y^{2} d y

\int_{0}^{π} x^{2} cos (\frac{x}{2}) d x

\int_{- 2}^{2} (∣ x ∣) - (x^{2} - 2) d x

\int_{0}^{2 x} cos (x^{2}) d x

What is the integral from 0 to 1/2 of xcos((pikx)/2) ?
The integral from 0 to 1/2 of xcos((pikx)/2) is (piksin((pik)/4)+4cos((pik)/4)-4)/(pi^2k^2)