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

The integral of sin^3(x)sqrt(cos(x)) is -2/3 cos^{3/2}(x)+2/7 cos^{7/2}(x)+C

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integral of sin^3(x)sqrt(cos(x))

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

- \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x) + C

Solution steps

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

Apply u-substitution

: \int - u (1 - u^{2}) d u

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

Take the constant out:

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

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

Expand

u (1 - u^{2}) : u - u^{\frac{5}{2}}

= - \int u - u^{\frac{5}{2}} d u

Apply the Sum Rule:

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

= - (\int u d u - \int u^{\frac{5}{2}} d u)

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

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

= - (\frac{2}{3} u^{\frac{3}{2}} - \frac{2}{7} u^{\frac{7}{2}})

Substitute back

u = cos (x)

= - (\frac{2}{3} cos^{\frac{3}{2}} (x) - \frac{2}{7} cos^{\frac{7}{2}} (x))

Simplify

= - \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x)

Add a constant to the solution

= - \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x) + C

\int 15 cos^{3} (2 x) sin^{5} (2 x) d x

x - 2 sin (x)

\frac{d}{d x} (4 x^{2} \cdot 2^{(3 x + 1)})

What is the integral of sin^3(x)sqrt(cos(x)) ?
The integral of sin^3(x)sqrt(cos(x)) is -2/3 cos^{3/2}(x)+2/7 cos^{7/2}(x)+C