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(1+cos(x))/x^2
Solving integrals/ (1+cos(x))/x^2

Integral of (1+cos(x))/x^2 dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1/2             
   /              
  |               
  |  1 + cos(x)   
  |  ---------- dx
  |       2       
  |      x        
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 /                
3/10
∫31012cos⁡(x)+1x2 dx\int\limits_{\frac{3}{10}}^{\frac{1}{2}} \frac{\cos{\left(x \right)} + 1}{x^{2}}\, dx 
Integral((1 + cos(x))/(x^2), (x, 3/10, 1/2))
Detail solution

  1. Rewrite the integrand:

    cos⁡(x)+1x2=cos⁡(x)x2+1x2\frac{\cos{\left(x \right)} + 1}{x^{2}} = \frac{\cos{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}

  2. Integrate term-by-term:

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

      But the integral is

      −Si⁡(x)−cos⁡(x)x- \operatorname{Si}{\left(x \right)} - \frac{\cos{\left(x \right)}}{x}

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫1x2 dx=−1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

    The result is: −Si⁡(x)−cos⁡(x)x−1x- \operatorname{Si}{\left(x \right)} - \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}

  3. Now simplify:

    −xSi⁡(x)+cos⁡(x)+1x- \frac{x \operatorname{Si}{\left(x \right)} + \cos{\left(x \right)} + 1}{x}

  4. Add the constant of integration:

    −xSi⁡(x)+cos⁡(x)+1x+constant- \frac{x \operatorname{Si}{\left(x \right)} + \cos{\left(x \right)} + 1}{x}+ \mathrm{constant}

The answer is:

−xSi⁡(x)+cos⁡(x)+1x+constant- \frac{x \operatorname{Si}{\left(x \right)} + \cos{\left(x \right)} + 1}{x}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                      
 |                                       
 | 1 + cos(x)          1           cos(x)
 | ---------- dx = C - - - Si(x) - ------
 |      2              x             x   
 |     x                                 
 |                                       
/
−1x−i Γ(−1,i x)−i Γ(−1,−i x)2-{{1}\over{x}}-{{i\,\Gamma\left(-1 , i\,x\right)-i\,\Gamma\left(-1 , -i\,x\right)}\over{2}}
Simplify
The graph
0.300.500.320.340.360.380.400.420.440.460.48-5050
Plot the graph f(x)
The answer [src] 
4                          10*cos(3/10)           
- - Si(1/2) - 2*cos(1/2) + ------------ + Si(3/10)
3                               3
−3 i Γ(−1,i2)−3 i Γ(−1,3 i10)+3 i Γ(−1,−3 i10)−3 i Γ(−1,−i2)−86-{{3\,i\,\Gamma\left(-1 , {{i}\over{2}}\right)-3\,i\,\Gamma\left(-1 , {{3\,i}\over{10}}\right)+3\,i\,\Gamma\left(-1 , -{{3\,i}\over{10 }}\right)-3\,i\,\Gamma\left(-1 , -{{i}\over{2}}\right)-8}\over{6}} 
=
= 
4                          10*cos(3/10)           
- - Si(1/2) - 2*cos(1/2) + ------------ + Si(3/10)
3                               3
−2cos⁡(12)−Si⁡(12)+Si⁡(310)+43+10cos⁡(310)3- 2 \cos{\left(\frac{1}{2} \right)} - \operatorname{Si}{\left(\frac{1}{2} \right)} + \operatorname{Si}{\left(\frac{3}{10} \right)} + \frac{4}{3} + \frac{10 \cos{\left(\frac{3}{10} \right)}}{3}
Simplify
Numerical answer [src] 
2.56801979906858
2.56801979906858
The graph
Integral of (1+cos(x))/x^2 dx

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

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