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

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

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integral of cos(3/2 x)

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

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

Solution steps

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Apply u-substitution

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

= \int cos (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 cos (u) d u

Use the common integral:

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

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

Substitute back

u = \frac{3 x}{2}

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

Add a constant to the solution

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

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