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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

= \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 = sin (x)

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

Add a constant to the solution

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

y = \frac{a ^{4}}{a ^{3} - 8 x}

\frac{\partial}{\partial z} (z y^{2} (t + 1))

\int \frac{sec ( y )}{sin ( y )} d y

\frac{\partial}{\partial x} (\frac{1}{x ^{2}})

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