What is the integral from-1 to 3 of cos(sqrt(x+1)) ?

The integral from-1 to 3 of cos(sqrt(x+1)) is 2(2sin(2)+cos(2)-1)

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integral from-1 to 3 of cos(sqrt(x+1))

\int_{- 1}^{3} cos (x + 1) d x

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

Decimal

0.80489 \dots

Solution steps

\int_{- 1}^{3} cos (x + 1) d x

Apply u-substitution

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

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

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

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

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What is the integral from-1 to 3 of cos(sqrt(x+1)) ?
The integral from-1 to 3 of cos(sqrt(x+1)) is 2(2sin(2)+cos(2)-1)