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Solving integrals/ sin(x)*sqrt(1-cos(x))

Integral of sin(x)*sqrt(1-cos(x)) dx

The solution

You have entered [src]

  1                         
  /                         
 |                          
 |           ____________   
 |  sin(x)*\/ 1 - cos(x)  dx
 |                          
/                           
0

$$\int\limits_{0}^{1} \sqrt{1 - \cos{\left(x \right)}} \sin{\left(x \right)}\, dx$$

Integral(sin(x)*sqrt(1 - cos(x)), (x, 0, 1))

Detail solution

Let $u = 1 - \cos{\left(x \right)}$ .

Then let $du = \sin{\left(x \right)} dx$ and substitute $du$ :

$\int \sqrt{u}\, du$
1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

$\frac{2 \left(1 - \cos{\left(x \right)}\right)^{\frac{3}{2}}}{3}$
Add the constant of integration:

$\frac{2 \left(1 - \cos{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(1 - \cos{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                
 |                                              3/2
 |          ____________          2*(1 - cos(x))   
 | sin(x)*\/ 1 - cos(x)  dx = C + -----------------
 |                                        3        
/

$$\int \sqrt{1 - \cos{\left(x \right)}} \sin{\left(x \right)}\, dx = C + \frac{2 \left(1 - \cos{\left(x \right)}\right)^{\frac{3}{2}}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ____________       ____________       
2*\/ 1 - cos(1)    2*\/ 1 - cos(1) *cos(1)
---------------- - -----------------------
       3                      3

$$- \frac{2 \sqrt{1 - \cos{\left(1 \right)}} \cos{\left(1 \right)}}{3} + \frac{2 \sqrt{1 - \cos{\left(1 \right)}}}{3}$$

    ____________       ____________       
2*\/ 1 - cos(1)    2*\/ 1 - cos(1) *cos(1)
---------------- - -----------------------
       3                      3

$$- \frac{2 \sqrt{1 - \cos{\left(1 \right)}} \cos{\left(1 \right)}}{3} + \frac{2 \sqrt{1 - \cos{\left(1 \right)}}}{3}$$

2*sqrt(1 - cos(1))/3 - 2*sqrt(1 - cos(1))*cos(1)/3

Numerical answer [src]

0.207786452690549

0.207786452690549

Use the examples entering the upper and lower limits of integration.

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Integral of sin(x)*sqrt(1-cos(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution