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

The integral from 0 to 1/2 of cos(sqrt(x)) is 2(sqrt(1/2)sin(sqrt(1/2))+cos(sqrt(1/2))-1)

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integral from 0 to 1/2 of cos(sqrt(x))

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

2 (\frac{1}{2} sin (\frac{1}{2}) + cos (\frac{1}{2}) - 1)

Decimal

0.43921 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{\frac{1}{2}} cos (u) \cdot 2 u d u

= \int_{0}^{\frac{1}{2}} 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}^{\frac{1}{2}} cos (u) u d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0}^{\frac{1}{2}}

= 2 [u sin (u) - \int sin (u) d u]_{0}^{\frac{1}{2}}

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

= 2 [u sin (u) - (- cos (u))]_{0}^{\frac{1}{2}}

Simplify

= 2 [u sin (u) + cos (u)]_{0}^{\frac{1}{2}}

Compute the boundaries:

\frac{1}{2} sin (\frac{1}{2}) + cos (\frac{1}{2}) - 1

= 2 (\frac{1}{2} sin (\frac{1}{2}) + cos (\frac{1}{2}) - 1)

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What is the integral from 0 to 1/2 of cos(sqrt(x)) ?
The integral from 0 to 1/2 of cos(sqrt(x)) is 2(sqrt(1/2)sin(sqrt(1/2))+cos(sqrt(1/2))-1)