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Solving integrals/ sinx/(3+2cosx)½

Integral of sinx/(3+2cosx)½ dx

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The solution

You have entered [src] 
  1                  
  /                  
 |                   
 |  /   sin(x)   \   
 |  |------------|   
 |  \3 + 2*cos(x)/   
 |  -------------- dx
 |        2          
 |                   
/                    
0
∫0112cos⁡(x)+3sin⁡(x)2 dx\int\limits_{0}^{1} \frac{\frac{1}{2 \cos{\left(x \right)} + 3} \sin{\left(x \right)}}{2}\, dx 
Integral((sin(x)/(3 + 2*cos(x)))/2, (x, 0, 1))
Detail solution

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

    ∫12cos⁡(x)+3sin⁡(x)2 dx=∫sin⁡(x)2cos⁡(x)+3 dx2\int \frac{\frac{1}{2 \cos{\left(x \right)} + 3} \sin{\left(x \right)}}{2}\, dx = \frac{\int \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)} + 3}\, dx}{2}

    1. Let u=2cos⁡(x)+3u = 2 \cos{\left(x \right)} + 3.

      Then let du=−2sin⁡(x)dxdu = - 2 \sin{\left(x \right)} dx and substitute −du2- \frac{du}{2}:

      ∫(−12u) du\int \left(- \frac{1}{2 u}\right)\, du

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

        ∫1u du=−∫1u du2\int \frac{1}{u}\, du = - \frac{\int \frac{1}{u}\, du}{2}

        1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

        So, the result is: −log⁡(u)2- \frac{\log{\left(u \right)}}{2}

      Now substitute uu back in:

      −log⁡(2cos⁡(x)+3)2- \frac{\log{\left(2 \cos{\left(x \right)} + 3 \right)}}{2}

    So, the result is: −log⁡(2cos⁡(x)+3)4- \frac{\log{\left(2 \cos{\left(x \right)} + 3 \right)}}{4}

  2. Add the constant of integration:

    −log⁡(2cos⁡(x)+3)4+constant- \frac{\log{\left(2 \cos{\left(x \right)} + 3 \right)}}{4}+ \mathrm{constant}

The answer is:

−log⁡(2cos⁡(x)+3)4+constant- \frac{\log{\left(2 \cos{\left(x \right)} + 3 \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                         
 |                                          
 | /   sin(x)   \                           
 | |------------|                           
 | \3 + 2*cos(x)/          log(3 + 2*cos(x))
 | -------------- dx = C - -----------------
 |       2                         4        
 |                                          
/
∫12cos⁡(x)+3sin⁡(x)2 dx=C−log⁡(2cos⁡(x)+3)4\int \frac{\frac{1}{2 \cos{\left(x \right)} + 3} \sin{\left(x \right)}}{2}\, dx = C - \frac{\log{\left(2 \cos{\left(x \right)} + 3 \right)}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.5-0.5
Plot the graph f(x)
The answer [src] 
  log(3/2 + cos(1))   log(5/2)
- ----------------- + --------
          4              4
−log⁡(cos⁡(1)+32)4+log⁡(52)4- \frac{\log{\left(\cos{\left(1 \right)} + \frac{3}{2} \right)}}{4} + \frac{\log{\left(\frac{5}{2} \right)}}{4} 
=
= 
  log(3/2 + cos(1))   log(5/2)
- ----------------- + --------
          4              4
−log⁡(cos⁡(1)+32)4+log⁡(52)4- \frac{\log{\left(\cos{\left(1 \right)} + \frac{3}{2} \right)}}{4} + \frac{\log{\left(\frac{5}{2} \right)}}{4} 
-log(3/2 + cos(1))/4 + log(5/2)/4
Numerical answer [src] 
0.0507981864614772
0.0507981864614772

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

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