What is the integral of 5sin(x/2) ?

The integral of 5sin(x/2) is -10cos(x/2)+C

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integral of 5sin(x/2)

\int 5 sin (\frac{x}{2}) d x

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

Solution steps

\int 5 sin (\frac{x}{2}) d x

Take the constant out:

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

= 5 \cdot \int sin (\frac{x}{2}) d x

Apply u-substitution

= 5 \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

= 5 \cdot 2 \cdot \int sin (u) d u

Use the common integral:

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

= 5 \cdot 2 (- cos (u))

Substitute back

u = \frac{x}{2}

= 5 \cdot 2 (- cos (\frac{x}{2}))

Simplify

= - 10 cos (\frac{x}{2})

Add a constant to the solution

= - 10 cos (\frac{x}{2}) + C