What is the integral from 0 to 0.5 of sin(sqrt(x)) ?

The integral from 0 to 0.5 of sin(sqrt(x)) is 0.22412…

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integral from 0 to 0.5 of sin(sqrt(x))

\int_{0}^{0.5} sin (x) d x

0.22412 \dots

Solution steps

\int_{0}^{0.5} sin (x) d x

Apply u-substitution

: \int_{0}^{0.70710 \dots} sin (u) \cdot 2 u d u

= \int_{0}^{0.70710 \dots} 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}^{0.70710 \dots} sin (u) u d u

Apply Integration By Parts

: [- u cos (u) - \int - cos (u) d u]_{0}^{0.70710 \dots}

= 2 [- u cos (u) - \int - cos (u) d u]_{0}^{0.70710 \dots}

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

= 2 [- u cos (u) - (- sin (u))]_{0}^{0.70710 \dots}

Simplify

= 2 [- u cos (u) + sin (u)]_{0}^{0.70710 \dots}

Compute the boundaries:

0.11206 \dots

= 2 \cdot 0.11206 \dots

Simplify

= 0.22412 \dots

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What is the integral from 0 to 0.5 of sin(sqrt(x)) ?
The integral from 0 to 0.5 of sin(sqrt(x)) is 0.22412…