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Identical expressions
sqrt(sin2x+ one)cos2x
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sqrt(sin2x+1)cos2xdx
Similar expressions
sqrt(sin2x-1)cos2x

Integral of sqrt(sin2x+1)cos2x dx

The solution

You have entered [src]

  1                             
  /                             
 |                              
 |    ______________            
 |  \/ sin(2*x) + 1 *cos(2*x) dx
 |                              
/                               
0

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sin{\left(2 x \right)} + 1$ .
  
  Then let $du = 2 \cos{\left(2 x \right)} dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{\sqrt{u}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{2}$
    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: $\frac{u^{\frac{3}{2}}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\left(\sin{\left(2 x \right)} + 1\right)^{\frac{3}{2}}}{3}$
Method #2
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{\sqrt{\sin{\left(u \right)} + 1} \cos{\left(u \right)}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sqrt{\sin{\left(u \right)} + 1} \cos{\left(u \right)}\, du = \frac{\int \sqrt{\sin{\left(u \right)} + 1} \cos{\left(u \right)}\, du}{2}$
    1. Let $u = \sin{\left(u \right)} + 1$ .
      
      Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
      
      $\int \sqrt{u}\, du$
      
      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:
      
      $\frac{2 \left(\sin{\left(u \right)} + 1\right)^{\frac{3}{2}}}{3}$
    So, the result is: $\frac{\left(\sin{\left(u \right)} + 1\right)^{\frac{3}{2}}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\left(\sin{\left(2 x \right)} + 1\right)^{\frac{3}{2}}}{3}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

        ____________     ____________       
  1   \/ 1 + sin(2)    \/ 1 + sin(2) *sin(2)
- - + -------------- + ---------------------
  3         3                    3

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

        ____________     ____________       
  1   \/ 1 + sin(2)    \/ 1 + sin(2) *sin(2)
- - + -------------- + ---------------------
  3         3                    3

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

-1/3 + sqrt(1 + sin(2))/3 + sqrt(1 + sin(2))*sin(2)/3

Numerical answer [src]

0.546072062781404

0.546072062781404

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

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Identical expressions

Similar expressions

Integral of sqrt(sin2x+1)cos2x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2