What is the integral of sqrt(x)cos(sqrt(x^3)) ?

The integral of sqrt(x)cos(sqrt(x^3)) is 2/3 sin(x^{3/2})+C

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integral of sqrt(x)cos(sqrt(x^3))

\int x cos (x^{3}) d x

\frac{2}{3} sin (x^{\frac{3}{2}}) + C

Solution steps

\int x cos (x^{3}) d x

x^{3} = x^{\frac{3}{2}},

assuming

x \geq 0

= \int x cos (x^{\frac{3}{2}}) d x

Apply u-substitution

: \int 2 u^{2} cos (u^{3}) d u

= \int 2 u^{2} cos (u^{3}) 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^{3}) d u

Apply u-substitution

: \int \frac{cos ( v )}{3} d v

= 2 \cdot \int \frac{cos ( v )}{3} d v

Take the constant out:

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

= 2 \cdot \frac{1}{3} \cdot \int cos (v) d v

Use the common integral:

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

= 2 \cdot \frac{1}{3} sin (v)

Substitute back

= 2 \cdot \frac{1}{3} sin ((x)^{3})

Simplify

2 \cdot \frac{1}{3} sin ((x)^{3}) : \frac{2}{3} sin (x^{\frac{3}{2}})

= \frac{2}{3} sin (x^{\frac{3}{2}})

Add a constant to the solution

= \frac{2}{3} sin (x^{\frac{3}{2}}) + C

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What is the integral of sqrt(x)cos(sqrt(x^3)) ?
The integral of sqrt(x)cos(sqrt(x^3)) is 2/3 sin(x^{3/2})+C