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

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

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integral of 5cos(x/2)

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

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

Solution steps

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

Apply u-substitution

= 5 \cdot \int cos (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 cos (u) d u

Use the common integral:

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

= 5 \cdot 2 sin (u)

Substitute back

u = \frac{x}{2}

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

Simplify

= 10 sin (\frac{x}{2})

Add a constant to the solution

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