What is the integral of (2cos^5(x))/(sqrt(sin(x))) ?

The integral of (2cos^5(x))/(sqrt(sin(x))) is 2(2sqrt(sin(x))-4/5 sin^{5/2}(x)+2/9 sin^{9/2}(x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2cos^5(x))/(sqrt(sin(x)))

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

2 (2 sin (x) - \frac{4}{5} sin^{\frac{5}{2}} (x) + \frac{2}{9} sin^{\frac{9}{2}} (x)) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{cos ^{5} ( x )}{sin ( x )} d x

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

\frac{( 1 - u ^{2} ) ^{2}}{u} : \frac{1}{u} - 2 u^{\frac{3}{2}} + u^{\frac{7}{2}}

= 2 \cdot \int \frac{1}{u} - 2 u^{\frac{3}{2}} + u^{\frac{7}{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

= 2 (\int \frac{1}{u} d u - \int 2 u^{\frac{3}{2}} d u + \int u^{\frac{7}{2}} d u)

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

\int 2 u^{\frac{3}{2}} d u = \frac{4}{5} u^{\frac{5}{2}}

\int u^{\frac{7}{2}} d u = \frac{2}{9} u^{\frac{9}{2}}

= 2 (2 u - \frac{4}{5} u^{\frac{5}{2}} + \frac{2}{9} u^{\frac{9}{2}})

Substitute back

u = sin (x)

= 2 (2 sin (x) - \frac{4}{5} sin^{\frac{5}{2}} (x) + \frac{2}{9} sin^{\frac{9}{2}} (x))

Add a constant to the solution

= 2 (2 sin (x) - \frac{4}{5} sin^{\frac{5}{2}} (x) + \frac{2}{9} sin^{\frac{9}{2}} (x)) + C

\frac{d}{d x} (tan (2 x))

\frac{d}{d x} (\frac{cot ( x )}{sin ( x )})

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

x \to - 2 lim (\frac{- 9}{x ^{2}})

\frac{d}{d x} (8 x + 7)

What is the integral of (2cos^5(x))/(sqrt(sin(x))) ?
The integral of (2cos^5(x))/(sqrt(sin(x))) is 2(2sqrt(sin(x))-4/5 sin^{5/2}(x)+2/9 sin^{9/2}(x))+C