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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int 2 u^{2} sin (u^{3} + 1) d u

= \int 2 u^{2} sin (u^{3} + 1) 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^{3} + 1) d u

Apply u-substitution

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

= 2 \cdot \int \frac{sin ( 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 sin (v) d v

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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