What is the integral of sin(x^{1/2}) ?

The integral of sin(x^{1/2}) is 2(-x^{1/2}cos(x^{1/2})+sin(x^{1/2}))+C

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integral of sin(x^{1/2})

\int sin (x^{\frac{1}{2}}) d x

2 (- x^{\frac{1}{2}} cos (x^{\frac{1}{2}}) + sin (x^{\frac{1}{2}})) + C

Solution steps

\int sin (x^{\frac{1}{2}}) d x

Apply u-substitution

: \int sin (u) \cdot 2 u d u

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

Apply Integration By Parts

: - u cos (u) - \int - cos (u) d u

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

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

= 2 (- u cos (u) - (- sin (u)))

Substitute back

u = x^{\frac{1}{2}}

= 2 (- x^{\frac{1}{2}} cos (x^{\frac{1}{2}}) - (- sin (x^{\frac{1}{2}})))

Simplify

= 2 (- x^{\frac{1}{2}} cos (x^{\frac{1}{2}}) + sin (x^{\frac{1}{2}}))

Add a constant to the solution

= 2 (- x^{\frac{1}{2}} cos (x^{\frac{1}{2}}) + sin (x^{\frac{1}{2}})) + C

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x \frac{d y}{d x} + 3 y = x^{3} - x

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What is the integral of sin(x^{1/2}) ?
The integral of sin(x^{1/2}) is 2(-x^{1/2}cos(x^{1/2})+sin(x^{1/2}))+C