What is the integral of cos^3(x/(19)) ?

The integral of cos^3(x/(19)) is 19(sin(x/(19))-(sin^3(x/(19)))/3)+C

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integral of cos^3(x/(19))

\int cos^{3} (\frac{x}{19}) d x

19 (sin (\frac{x}{19}) - \frac{sin ^{3} ( \frac{x}{19} )}{3}) + C

Solution steps

\int cos^{3} (\frac{x}{19}) d x

Apply u-substitution

: \int cos^{3} (u) \cdot 19 d u

= \int cos^{3} (u) \cdot 19 d u

Take the constant out:

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

= 19 \cdot \int cos^{3} (u) d u

Rewrite using trig identities

= 19 \cdot \int (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

: \int 1 - v^{2} d v

= 19 \cdot \int 1 - v^{2} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 19 (\int 1 d v - \int v^{2} d v)

\int 1 d v = v

\int v^{2} d v = \frac{v ^{3}}{3}

= 19 (v - \frac{v ^{3}}{3})

Substitute back

= 19 (sin (\frac{x}{19}) - \frac{sin ^{3} ( \frac{x}{19} )}{3})

Add a constant to the solution

= 19 (sin (\frac{x}{19}) - \frac{sin ^{3} ( \frac{x}{19} )}{3}) + C

\int (cos (x))^{3} d x

\frac{\partial}{\partial y} (arctan (x + 2 y))

\int \frac{5 - 5 x}{1 - x} d x

x^{^{''}} + 8 x^{^{'}} + 16 x = 0

\frac{d}{d x} (5 x - 8)

What is the integral of cos^3(x/(19)) ?
The integral of cos^3(x/(19)) is 19(sin(x/(19))-(sin^3(x/(19)))/3)+C