What is the integral from 0 to 1 of 2cos((pix)/2) ?

The integral from 0 to 1 of 2cos((pix)/2) is 4/pi

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 of 2cos((pix)/2)

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

\frac{4}{π}

Decimal Notation

1.27323 \dots

Solution steps

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

Take the constant out:

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

= 2 \cdot \int_{0}^{1} cos (\frac{π x}{2}) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= 2 \cdot \frac{2}{π} [sin (u)]_{0}^{\frac{π}{2}}

Simplify

2 \cdot \frac{2}{π} [sin (u)]_{0}^{\frac{π}{2}} : \frac{4}{π} [sin (u)]_{0}^{\frac{π}{2}}

= \frac{4}{π} [sin (u)]_{0}^{\frac{π}{2}}

Compute the boundaries:

1

= \frac{4}{π} \cdot 1

Simplify

= \frac{4}{π}

\int_{0}^{2 π} 0 d x

\int_{0}^{1} (1 + x^{2})^{- \frac{3}{2}} d x

\int_{a}^{x} f (t) d t

\int_{4}^{7} 4 d x

\int_{0}^{a} (a^{2} - x^{2}) d x

What is the integral from 0 to 1 of 2cos((pix)/2) ?
The integral from 0 to 1 of 2cos((pix)/2) is 4/pi