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

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

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integral from 1 to 2 of cos(sqrt(x-1))

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

2 (sin (1) + cos (1) - 1)

Decimal Notation

0.76354 \dots

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

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

= \int_{0}^{1} cos (v) \cdot 2 v d v

Take the constant out:

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

= 2 \cdot \int_{0}^{1} cos (v) v d v

Apply Integration By Parts

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

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

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

= 2 [v sin (v) - (- cos (v))]_{0}^{1}

Simplify

= 2 [v sin (v) + cos (v)]_{0}^{1}

Compute the boundaries:

sin (1) + cos (1) - 1

= 2 (sin (1) + cos (1) - 1)

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