What is the integral of sqrt(x)sin(1+x^{3/2}) ?

The integral of sqrt(x)sin(1+x^{3/2}) is -2/3 cos(1+x^{3/2})+C

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integral of sqrt(x)sin(1+x^{3/2})

\int x sin (1 + x^{\frac{3}{2}}) d x

- \frac{2}{3} cos (1 + x^{\frac{3}{2}}) + C

Solution steps

\int x sin (1 + x^{\frac{3}{2}}) d x

Apply u-substitution

= \int \frac{2 sin ( u )}{3} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{2}{3} (- cos (u))

Substitute back

u = 1 + x^{\frac{3}{2}}

= \frac{2}{3} (- cos (1 + x^{\frac{3}{2}}))

Simplify

= - \frac{2}{3} cos (1 + x^{\frac{3}{2}})

Add a constant to the solution

= - \frac{2}{3} cos (1 + x^{\frac{3}{2}}) + C

What is the integral of sqrt(x)sin(1+x^{3/2}) ?
The integral of sqrt(x)sin(1+x^{3/2}) is -2/3 cos(1+x^{3/2})+C