What is the integral from 0.5 to 1 of cos(sqrt(x)) ?

The integral from 0.5 to 1 of cos(sqrt(x)) is 0.32433…

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integral from 0.5 to 1 of cos(sqrt(x))

\int_{0.5}^{1} cos (x) d x

0.32433 \dots

Solution steps

\int_{0.5}^{1} cos (x) d x

Apply u-substitution

: \int_{0.70710 \dots}^{1} cos (u) \cdot 2 u d u

= \int_{0.70710 \dots}^{1} cos (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_{0.70710 \dots}^{1} cos (u) u d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0.70710 \dots}^{1}

= 2 [u sin (u) - \int sin (u) d u]_{0.70710 \dots}^{1}

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

= 2 [u sin (u) - (- cos (u))]_{0.70710 \dots}^{1}

Simplify

= 2 [u sin (u) + cos (u)]_{0.70710 \dots}^{1}

Compute the boundaries:

0.16216 \dots

= 2 \cdot 0.16216 \dots

Simplify

= 0.32433 \dots

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What is the integral from 0.5 to 1 of cos(sqrt(x)) ?
The integral from 0.5 to 1 of cos(sqrt(x)) is 0.32433…