What is the integral from 0 to pi^2 of cos(sqrt(x)) ?

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

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integral from 0 to pi^2 of cos(sqrt(x))

\int_{0}^{π^{2}} cos (x) d x

- 4

Solution steps

\int_{0}^{π^{2}} cos (x) d x

Apply u-substitution

: \int_{0}^{π} cos (u) \cdot 2 u d u

= \int_{0}^{π} 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}^{π} cos (u) u d u

Apply Integration By Parts

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

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

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

= 2 [u sin (u) - (- cos (u))]_{0}^{π}

Simplify

= 2 [u sin (u) + cos (u)]_{0}^{π}

Compute the boundaries:

- 2

= 2 (- 2)

Simplify

= - 4

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