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

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

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integral of sqrt(x)sin(sqrt(x))

\int x sin (x) d x

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

Solution steps

\int x sin (x) d x

Apply u-substitution

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

Apply Integration By Parts

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

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

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

Substitute back

u = x

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

Simplify

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

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

Add a constant to the solution

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

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