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Identical expressions
sqrt(sin(three x))cos(3x)+3
square root of ( sinus of (3x)) co sinus of e of (3x) plus 3
square root of ( sinus of (three x)) co sinus of e of (3x) plus 3
√(sin(3x))cos(3x)+3
sqrtsin3xcos3x+3
sqrt(sin(3x))cos(3x)+3dx
Similar expressions
sqrt(sin(3x))cos(3x)-3

Solving integrals/ sqrt(sin(3x))cos(3x)+3

Integral of sqrt(sin(3x))cos(3x)+3 dx

The solution

You have entered [src]

  1                               
  /                               
 |                                
 |  /  __________             \   
 |  \\/ sin(3*x) *cos(3*x) + 3/ dx
 |                                
/                                 
0

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

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

Detail solution

Integrate term-by-term:
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \sin{\left(3 x \right)}$ .
    
    Then let $du = 3 \cos{\left(3 x \right)} dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{\sqrt{u}}{3}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{3}$
      
      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: $\frac{2 u^{\frac{3}{2}}}{9}$
    Now substitute $u$ back in:
    
    $\frac{2 \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}$
  Method #2
  1. Let $u = 3 x$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{\sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}}{3}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}\, du = \frac{\int \sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}\, du}{3}$
      
      Let $u = \sin{\left(u \right)}$ .
      
      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:
      
      $\frac{2 \sin^{\frac{3}{2}}{\left(u \right)}}{3}$
      
      So, the result is: $\frac{2 \sin^{\frac{3}{2}}{\left(u \right)}}{9}$
    Now substitute $u$ back in:
    
    $\frac{2 \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 3\, dx = 3 x$
The result is: $3 x + \frac{2 \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

         3/2   
    2*sin   (3)
3 + -----------
         9

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

         3/2   
    2*sin   (3)
3 + -----------
         9

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

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

Numerical answer [src]

3.01178068042735

3.01178068042735

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

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Identical expressions

Similar expressions

Integral of sqrt(sin(3x))cos(3x)+3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2