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Solving integrals/ cosxsinx*((cosx)^2-(sinx)^2)^(1/2)dx

Integral of cosxsinx*((cosx)^2-(sinx)^2)^(1/2)dx dx

The solution

You have entered [src]

  1                                          
  /                                          
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 |                   ___________________     
 |                  /    2         2         
 |  cos(x)*sin(x)*\/  cos (x) - sin (x) *1 dx
 |                                           
/                                            
0

$$\int\limits_{0}^{1} \cos{\left(x \right)} \sin{\left(x \right)} \sqrt{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 1\, dx$$

Integral(cos(x)*sin(x)*sqrt(cos(x)^2 - sin(x)^2)*1, (x, 0, 1))

Detail solution

Let $u = - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$ .

Then let $du = - 4 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $- \frac{du}{4}$ :

$\int \frac{\sqrt{u}}{16}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sqrt{u}}{4}\right)\, du = - \frac{\int \sqrt{u}\, du}{4}$
  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: $- \frac{u^{\frac{3}{2}}}{6}$
Now substitute $u$ back in:

$- \frac{\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)^{\frac{3}{2}}}{6}$
Now simplify:

$- \frac{\cos^{\frac{3}{2}}{\left(2 x \right)}}{6}$
Add the constant of integration:

$- \frac{\cos^{\frac{3}{2}}{\left(2 x \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{\cos^{\frac{3}{2}}{\left(2 x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                      
 |                                                                    3/2
 |                  ___________________            /   2         2   \   
 |                 /    2         2                \cos (x) - sin (x)/   
 | cos(x)*sin(x)*\/  cos (x) - sin (x) *1 dx = C - ----------------------
 |                                                           6           
/

$$-{{\left(\cos ^2x-\sin ^2x\right)^{{{3}\over{2}}}}\over{6}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       ___________________              ___________________        
      /    2         2        2        /    2         2        2   
1   \/  cos (1) - sin (1) *cos (1)   \/  cos (1) - sin (1) *sin (1)
- - ------------------------------ + ------------------------------
6                 6                                6

$${{\sqrt{\cos ^21-\sin ^21}\,\left(\sin ^21-\cos ^21\right)}\over{6 }}+{{1}\over{6}}$$

       ___________________              ___________________        
      /    2         2        2        /    2         2        2   
1   \/  cos (1) - sin (1) *cos (1)   \/  cos (1) - sin (1) *sin (1)
- - ------------------------------ + ------------------------------
6                 6                                6

$$\frac{1}{6} - \frac{\sqrt{- \sin^{2}{\left(1 \right)} + \cos^{2}{\left(1 \right)}} \cos^{2}{\left(1 \right)}}{6} + \frac{\sqrt{- \sin^{2}{\left(1 \right)} + \cos^{2}{\left(1 \right)}} \sin^{2}{\left(1 \right)}}{6}$$

Simplify

Numerical answer [src]

(0.166644691984575 + 0.0447977230063355j)

(0.166644691984575 + 0.0447977230063355j)

The graph

Integral of cosxsinx*((cosx)^2-(sinx)^2)^(1/2)dx dx

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

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Identical expressions

Similar expressions

Integral of cosxsinx*((cosx)^2-(sinx)^2)^(1/2)dx dx

Limits of integration:

The graph:

Piecewise:

The solution