What is the integral from 0 to pi/4 of sin(sqrt(x)) ?

The integral from 0 to pi/4 of sin(sqrt(x)) is 2sin((sqrt(pi))/2)-sqrt(pi)cos((sqrt(pi))/2)

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integral from 0 to pi/4 of sin(sqrt(x))

\int_{0}^{\frac{π}{4}} sin (x) d x

2 sin (\frac{π}{2}) - π cos (\frac{π}{2})

Decimal Notation

0.42859 \dots

Solution steps

\int_{0}^{\frac{π}{4}} sin (x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

: [- u cos (u) - \int - cos (u) d u]_{0}^{\frac{π}{2}}

= 2 [- u cos (u) - \int - cos (u) d u]_{0}^{\frac{π}{2}}

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

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

Simplify

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

Compute the boundaries:

sin (\frac{π}{2}) - \frac{π}{2} cos (\frac{π}{2})

= 2 (sin (\frac{π}{2}) - \frac{π}{2} cos (\frac{π}{2}))

Simplify

= 2 sin (\frac{π}{2}) - π cos (\frac{π}{2})

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What is the integral from 0 to pi/4 of sin(sqrt(x)) ?
The integral from 0 to pi/4 of sin(sqrt(x)) is 2sin((sqrt(pi))/2)-sqrt(pi)cos((sqrt(pi))/2)