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

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

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integral of sqrt(x)cos(sqrt(x))

\int x cos (x) d x

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

Solution steps

\int x cos (x) d x

Apply u-substitution

= \int 2 u^{2} cos (u) d u

Take the constant out:

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

= 2 \cdot \int u^{2} cos (u) d u

Apply Integration By Parts

= 2 (u^{2} sin (u) - \int 2 u sin (u) d u)

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= 2 (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

Substitute back

u = x

= 2 ((x)^{2} sin (x) - 2 (- x cos (x) + sin (x)))

(x)^{2} = x

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

Add a constant to the solution

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

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