What is the integral of cos^3(x/(20)) ?

The integral of cos^3(x/(20)) is 20(sin(x/(20))-(sin^3(x/(20)))/3)+C

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integral of cos^3(x/(20))

\int cos^{3} (\frac{x}{20}) d x

20 (sin (\frac{x}{20}) - \frac{sin ^{3} ( \frac{x}{20} )}{3}) + C

Solution steps

\int cos^{3} (\frac{x}{20}) d x

Apply u-substitution

: \int cos^{3} (u) \cdot 20 d u

= \int cos^{3} (u) \cdot 20 d u

Take the constant out:

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

= 20 \cdot \int cos^{3} (u) d u

Rewrite using trig identities

= 20 \cdot \int (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

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

= 20 \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

= 20 (\int 1 d v - \int v^{2} d v)

\int 1 d v = v

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

= 20 (v - \frac{v ^{3}}{3})

Substitute back

= 20 (sin (\frac{x}{20}) - \frac{sin ^{3} ( \frac{x}{20} )}{3})

Add a constant to the solution

= 20 (sin (\frac{x}{20}) - \frac{sin ^{3} ( \frac{x}{20} )}{3}) + C

y^{^{''}} - 4 y^{^{'}} = 0

\frac{x e ^{t}}{1 + x e ^{t}}

h \to \infty lim (\frac{2}{h} + 3)

4 x^{4} - 4 x^{2}, 5 x^{2}, 0, 1.5

What is the integral of cos^3(x/(20)) ?
The integral of cos^3(x/(20)) is 20(sin(x/(20))-(sin^3(x/(20)))/3)+C