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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int - \frac{1 - u ^{2}}{u} d u

= \int - \frac{1 - u ^{2}}{u} d u

Expand

- \frac{1 - u ^{2}}{u} : - \frac{1}{u} + u^{\frac{3}{2}}

= \int - \frac{1}{u} + u^{\frac{3}{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 \frac{1}{u} d u + \int u^{\frac{3}{2}} d u

\int \frac{1}{u} d u = 2 u

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

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

Substitute back

u = cos (x)

= - 2 cos (x) + \frac{2}{5} cos^{\frac{5}{2}} (x)

Add a constant to the solution

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

\int (x^{2} + 5 x + 1) d x

\int (2 x + 1 - \frac{1}{x + 2}) d x

\int tan (10 x) sec^{2} (10 x) d x

\int \frac{5 x - 2}{( x - 1 ) ^{2} ( 2 x + 1 )} d x

\int \frac{1}{4 x ln ( x ) - 3 x} d x

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