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

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

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integral of cos(sqrt(x-1))

\int cos (x - 1) d x

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

Solution steps

\int cos (x - 1) d x

Apply u-substitution

: \int cos (u) d u

= \int cos (u) d u

Apply u-substitution

: \int cos (v) \cdot 2 v d v

= \int 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 cos (v) v d v

Apply Integration By Parts

: v sin (v) - \int sin (v) d v

= 2 (v sin (v) - \int sin (v) d v)

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

= 2 (v sin (v) - (- cos (v)))

Substitute back

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

Simplify

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

Add a constant to the solution

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

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\int cos^{2} (z)

\int 2 x e^{x y} + x^{2} y e^{x y} d x

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