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Integral of sin2x+1/sin2*x/3 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                           
  /                           
 |                            
 |  /               1     \   
 |  |sin(2*x) + ----------| dx
 |  \           sin(2*x)*3/   
 |                            
/                             
0                             
$$\int\limits_{0}^{1} \left(\sin{\left(2 x \right)} + \frac{1}{3 \sin{\left(2 x \right)}}\right)\, dx$$
Integral(sin(2*x) + 1/(sin(2*x)*3), (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

    1. There are multiple ways to do this integral.

      Method #1

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of sine is negative cosine:

          So, the result is:

        Now substitute back in:

      Method #2

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. There are multiple ways to do this integral.

          Method #1

          1. Let .

            Then let and substitute :

            1. The integral of a constant times a function is the constant times the integral of the function:

              1. The integral of is when :

              So, the result is:

            Now substitute back in:

          Method #2

          1. Let .

            Then let and substitute :

            1. The integral of is when :

            Now substitute back in:

        So, the result is:

    1. The integral of a constant times a function is the constant times the integral of the function:

      1. Don't know the steps in finding this integral.

        But the integral is

      So, the result is:

    The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                  
 |                                                                                   
 | /               1     \          cos(2*x)   log(1 + cos(2*x))   log(-1 + cos(2*x))
 | |sin(2*x) + ----------| dx = C - -------- - ----------------- + ------------------
 | \           sin(2*x)*3/             2               12                  12        
 |                                                                                   
/                                                                                    
$$\int \left(\sin{\left(2 x \right)} + \frac{1}{3 \sin{\left(2 x \right)}}\right)\, dx = C + \frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)}}{12} - \frac{\log{\left(\cos{\left(2 x \right)} + 1 \right)}}{12} - \frac{\cos{\left(2 x \right)}}{2}$$
The graph
The answer [src]
     pi*I
oo + ----
      12 
$$\infty + \frac{i \pi}{12}$$
=
=
     pi*I
oo + ----
      12 
$$\infty + \frac{i \pi}{12}$$
oo + pi*i/12
Numerical answer [src]
8.13031822795854
8.13031822795854

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