What is the integral of sin^4(x/3)cos(x/3) ?

The integral of sin^4(x/3)cos(x/3) is 3/5 sin^5(x/3)+C

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integral of sin^4(x/3)cos(x/3)

\int sin^{4} (\frac{x}{3}) cos (\frac{x}{3}) d x

\frac{3}{5} sin^{5} (\frac{x}{3}) + C

Solution steps

\int sin^{4} (\frac{x}{3}) cos (\frac{x}{3}) d x

Apply u-substitution

: \int sin^{4} (u) cos (u) \cdot 3 d u

= \int sin^{4} (u) cos (u) \cdot 3 d u

Take the constant out:

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

= 3 \cdot \int sin^{4} (u) cos (u) d u

Apply u-substitution

: \int v^{4} d v

= 3 \cdot \int v^{4} d v

Apply the Power Rule

: \frac{v ^{5}}{5}

= 3 \cdot \frac{v ^{5}}{5}

Substitute back

= 3 \cdot \frac{sin ^{5} ( \frac{x}{3} )}{5}

Simplify

3 \cdot \frac{sin ^{5} ( \frac{x}{3} )}{5} : \frac{3}{5} sin^{5} (\frac{x}{3})

= \frac{3}{5} sin^{5} (\frac{x}{3})

Add a constant to the solution

= \frac{3}{5} sin^{5} (\frac{x}{3}) + C

y^{5} = x^{6}

\frac{d}{d x} (\frac{- 0.4}{5 + 5 x})

\frac{d y}{d x} + 0.1 y = 10

\int_{1}^{2} \frac{3 ^{l n (x)}}{x} d x

\int 2^{3 x} d x

What is the integral of sin^4(x/3)cos(x/3) ?
The integral of sin^4(x/3)cos(x/3) is 3/5 sin^5(x/3)+C