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

The integral of (sqrt(sin^3(x)))/(sec(x)) is 2/5 sin^{5/2}(x)+C

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integral of (sqrt(sin^3(x)))/(sec(x))

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

\frac{2}{5} sin^{\frac{5}{2}} (x) + C

Solution steps

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

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

assuming

sin (x) \geq 0

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

Use the basic trigonometric identity:

\frac{1}{sec ( x )} = cos (x)

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

Apply u-substitution

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

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

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = sin (x)

= \frac{2}{5} sin^{\frac{5}{2}} (x)

Add a constant to the solution

= \frac{2}{5} sin^{\frac{5}{2}} (x) + C

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What is the integral of (sqrt(sin^3(x)))/(sec(x)) ?
The integral of (sqrt(sin^3(x)))/(sec(x)) is 2/5 sin^{5/2}(x)+C