What is the integral of sin^{1/2}(x)cos(x) ?

The integral of sin^{1/2}(x)cos(x) is 2/3 sin^{3/2}(x)+C

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integral of sin^{1/2}(x)cos(x)

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

\frac{2}{3} sin^{\frac{3}{2}} (x) + C

Solution steps

\int sin (x) cos (x) d x

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

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

Substitute back

u = sin (x)

= \frac{2}{3} sin^{\frac{3}{2}} (x)

Add a constant to the solution

= \frac{2}{3} sin^{\frac{3}{2}} (x) + C

x^{3} \frac{d y}{d x} + 3 x^{2} y = x

(\frac{2}{5} x)^{3}

\frac{\partial}{\partial x} (- 2 x^{2} + 4 y^{2} + 3 z^{2} + 43)

\frac{\partial}{\partial y} (2 x + y e^{x})

\frac{d}{d x} (- 4 x^{3} + 10 x^{2} - x)

What is the integral of sin^{1/2}(x)cos(x) ?
The integral of sin^{1/2}(x)cos(x) is 2/3 sin^{3/2}(x)+C