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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Apply Integration By Parts

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

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

= 2 [u sin (u) - (- cos (u))]_{x}^{1}

Simplify

= 2 [u sin (u) + cos (u)]_{x}^{1}

Compute the boundaries:

sin (1) + cos (1) - x sin (x) - cos (x)

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

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