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

The integral of 3sin(x/2) is -6cos(x/2)+C

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integral of 3sin(x/2)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int sin (u) \cdot 2 d u

= 3 \cdot \int sin (u) \cdot 2 d u

Take the constant out:

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

= 3 \cdot 2 \cdot \int sin (u) d u

Use the common integral:

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

= 3 \cdot 2 (- cos (u))

Substitute back

u = \frac{x}{2}

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

Simplify

= - 6 cos (\frac{x}{2})

Add a constant to the solution

= - 6 cos (\frac{x}{2}) + C

\int_{1}^{2} (4 x^{3} - 3 x^{2} + 2 x) d x

\int cos (3 x - 1) d x

\int (3 t e^{t}, ln (t)) d t

\int \frac{3 x ^{2}}{2} - x - 16.5 d x

2 x^{3} \frac{d y}{d x} = y (2 x^{2} - y^{2})