What is the integral of sin((3x)/2) ?

The integral of sin((3x)/2) is -2/3 cos((3x)/2)+C

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integral of sin((3x)/2)

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

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

Solution steps

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

Apply u-substitution

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

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

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

Simplify

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

Add a constant to the solution

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

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\int_{1}^{\infty} f (x) d x

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