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Identical expressions
sqrt(2sin3x)cos3x
square root of (2 sinus of 3x) co sinus of e of 3x
√(2sin3x)cos3x
sqrt2sin3xcos3x
sqrt(2sin3x)cos3xdx

Integral of sqrt(2sin3x)cos3x dx

The solution

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  1                           
  /                           
 |                            
 |    ____________            
 |  \/ 2*sin(3*x) *cos(3*x) dx
 |                            
/                             
0

$$\int\limits_{0}^{1} \sqrt{2 \sin{\left(3 x \right)}} \cos{\left(3 x \right)}\, dx$$

Integral(sqrt(2*sin(3*x))*cos(3*x), (x, 0, 1))

Detail solution

Let $u = 3 x$ .

Then let $du = 3 dx$ and substitute $\frac{\sqrt{2} du}{3}$ :

$\int \frac{\sqrt{2} \sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}}{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}\, du = \frac{\sqrt{2} \int \sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}\, du}{3}$
  1. Let $u = \sin{\left(u \right)}$ .
    
    Then let $du = \cos{\left(u \right)} du$ 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:
    
    $\frac{2 \sin^{\frac{3}{2}}{\left(u \right)}}{3}$
  So, the result is: $\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(u \right)}}{9}$
Now substitute $u$ back in:

$\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}$
Add the constant of integration:

$\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}+ \mathrm{constant}$

The answer is:

$\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
 |                                      ___    3/2     
 |   ____________                   2*\/ 2 *sin   (3*x)
 | \/ 2*sin(3*x) *cos(3*x) dx = C + -------------------
 |                                           9         
/

$$\int \sqrt{2 \sin{\left(3 x \right)}} \cos{\left(3 x \right)}\, dx = C + \frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___    3/2   
2*\/ 2 *sin   (3)
-----------------
        9

$$\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 \right)}}{9}$$

    ___    3/2   
2*\/ 2 *sin   (3)
-----------------
        9

$$\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 \right)}}{9}$$

2*sqrt(2)*sin(3)^(3/2)/9

Numerical answer [src]

0.0166603980343461

0.0166603980343461

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

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Identical expressions

Integral of sqrt(2sin3x)cos3x dx

Limits of integration:

The graph:

Piecewise:

The solution