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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

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

Solution steps

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

Apply u-substitution

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

Apply u-substitution

= 2 \cdot \int \frac{sin ( v )}{6} d v

Take the constant out:

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

= 2 \cdot \frac{1}{6} \cdot \int sin (v) d v

Use the common integral:

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

= 2 \cdot \frac{1}{6} (- cos (v))

Substitute back

= 2 \cdot \frac{1}{6} (- cos (2 (x)^{3}))

Simplify

2 \cdot \frac{1}{6} (- cos (2 (x)^{3})) : - \frac{1}{3} cos (2 x^{\frac{3}{2}})

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

Add a constant to the solution

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

in t e g r a l \frac{1}{( 1 - x )}

\int x x^{2} d x

\int - 2 e^{- x^{2}} d x

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

\int e^{e^{3 x} + 3 x} d x

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