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(cosx)^(one / two)sinx
( co sinus of e of x) to the power of (1 divide by 2) sinus of x
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(cosx)^(1/2)sinxdx

Integral of (cosx)^(1/2)sinx dx

The solution

You have entered [src]

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

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

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

Detail solution

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

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

$\int \left(- \sqrt{u}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = - \int \sqrt{u}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
  So, the result is: $- \frac{2 u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                      
 |                                 3/2   
 |   ________                 2*cos   (x)
 | \/ cos(x) *sin(x) dx = C - -----------
 |                                 3     
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

0.401899594820912

0.401899594820912

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

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Identical expressions

Integral of (cosx)^(1/2)sinx dx

Limits of integration:

The graph:

Piecewise:

The solution