Integral of sin3x/sinx dx

The solution

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  1            
  /            
 |             
 |  sin(3*x)   
 |  -------- dx
 |   sin(x)    
 |             
/              
0

\int\limits_{0}^{1} \frac{\sin{\left(3 x \right)}}{\sin{\left(x \right)}}\, dx

Integral(sin(3*x)/sin(x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{\sin{\left(3 x \right)}}{\sin{\left(x \right)}} = 3 - 4 \sin^{2}{\left(x \right)}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 3\, dx = 3 x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 4 \sin^{2}{\left(x \right)}\right)\, dx = - 4 \int \sin^{2}{\left(x \right)}\, dx$
  1. Rewrite the integrand:
    
    $\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
      1. Let $u = 2 x$ .
        
        Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du$
        
        The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
        
        Now substitute $u$ back in:
        
        $\frac{\sin{\left(2 x \right)}}{2}$
      So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$
    The result is: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
  So, the result is: $- 2 x + \sin{\left(2 x \right)}$
The result is: $x + \sin{\left(2 x \right)}$
Add the constant of integration:

$x + \sin{\left(2 x \right)}+ \mathrm{constant}$

The answer is:

x + \sin{\left(2 x \right)}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                              
 |                               
 | sin(3*x)                      
 | -------- dx = C + x + sin(2*x)
 |  sin(x)                       
 |                               
/

\int \frac{\sin{\left(3 x \right)}}{\sin{\left(x \right)}}\, dx = C + x + \sin{\left(2 x \right)}

Simplify

The graph

Plot the graph f(x)

The answer [src]

1 + sin(2)

\sin{\left(2 \right)} + 1

1 + sin(2)

\sin{\left(2 \right)} + 1

1 + sin(2)

Numerical answer [src]

1.90929742682568

1.90929742682568

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of sin3x/sinx dx

Limits of integration:

The graph:

Piecewise:

The solution