What is the integral of 18xcos(1/2 x) ?

The integral of 18xcos(1/2 x) is 72(1/2 xsin(x/2)+cos(x/2))+C

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integral of 18xcos(1/2 x)

\int 18 x cos (\frac{1}{2} x) d x

72 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2})) + C

Solution steps

\int 18 x cos (\frac{1}{2} x) d x

Take the constant out:

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

= 18 \cdot \int x cos (\frac{1}{2} x) d x

Simplify

\frac{1}{2} x : \frac{x}{2}

= 18 \cdot \int x cos (\frac{x}{2}) d x

Apply u-substitution

: \int 4 u cos (u) d u

= 18 \cdot \int 4 u cos (u) d u

Take the constant out:

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

= 18 \cdot 4 \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= 18 \cdot 4 (u sin (u) - \int sin (u) d u)

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

= 18 \cdot 4 (u sin (u) - (- cos (u)))

Substitute back

u = \frac{x}{2}

= 18 \cdot 4 (\frac{x}{2} sin (\frac{x}{2}) - (- cos (\frac{x}{2})))

Simplify

18 \cdot 4 (\frac{x}{2} sin (\frac{x}{2}) - (- cos (\frac{x}{2}))) : 72 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))

= 72 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))

Add a constant to the solution

= 72 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2})) + C

x \to 3 - lim (f (x) x)

\frac{\partial}{\partial x} (ln (x + c t))

(1 + \frac{2}{x})^{x}

\int \frac{3}{x ^{2} + 4} d x

- sin (2 t)

What is the integral of 18xcos(1/2 x) ?
The integral of 18xcos(1/2 x) is 72(1/2 xsin(x/2)+cos(x/2))+C