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xcosx/sin^3x
Solving integrals/ xcosx/sin^3x

Integral of xcosx/sin^3x dx

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The solution

You have entered [src] 
  0            
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 |  x*cos(x)   
 |  -------- dx
 |     3       
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∫00xcos⁡(x)sin⁡3(x) dx\int\limits_{0}^{0} \frac{x \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}\, dx
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Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=xu{\left(x \right)} = x and let dv⁡(x)=cos⁡(x)sin⁡3(x)\operatorname{dv}{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}.

    Then du⁡(x)=1\operatorname{du}{\left(x \right)} = 1.

    To find v(x)v{\left(x \right)}:

    1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

      Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

      ∫1u3 du\int \frac{1}{u^{3}}\, du

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫1u3 du=−12u2\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}

      Now substitute uu back in:

      −12sin⁡2(x)- \frac{1}{2 \sin^{2}{\left(x \right)}}

    Now evaluate the sub-integral.

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

    ∫(−12sin⁡2(x)) dx=−∫1sin⁡2(x) dx2\int \left(- \frac{1}{2 \sin^{2}{\left(x \right)}}\right)\, dx = - \frac{\int \frac{1}{\sin^{2}{\left(x \right)}}\, dx}{2}

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

      But the integral is

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

    So, the result is: cos⁡(x)2sin⁡(x)\frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)}}

  3. Now simplify:

    −x2sin⁡2(x)−12tan⁡(x)- \frac{x}{2 \sin^{2}{\left(x \right)}} - \frac{1}{2 \tan{\left(x \right)}}

  4. Add the constant of integration:

    −x2sin⁡2(x)−12tan⁡(x)+constant- \frac{x}{2 \sin^{2}{\left(x \right)}} - \frac{1}{2 \tan{\left(x \right)}}+ \mathrm{constant}

The answer is:

−x2sin⁡2(x)−12tan⁡(x)+constant- \frac{x}{2 \sin^{2}{\left(x \right)}} - \frac{1}{2 \tan{\left(x \right)}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                      
 |                                       
 | x*cos(x)              x        cos(x) 
 | -------- dx = C - --------- - --------
 |    3                   2      2*sin(x)
 | sin (x)           2*sin (x)           
 |                                       
/
(2 x sin⁡(2 x)−cos⁡(2 x)+1) sin⁡(4 x)+(sin⁡(2 x)+2 x cos⁡(2 x)) cos⁡(4 x)−4 x sin⁡2(2 x)−sin⁡(2 x)−4 x cos⁡2(2 x)+2 x cos⁡(2 x)sin⁡2(4 x)−4 sin⁡(2 x) sin⁡(4 x)+cos⁡2(4 x)+(2−4 cos⁡(2 x)) cos⁡(4 x)+4 sin⁡2(2 x)+4 cos⁡2(2 x)−4 cos⁡(2 x)+1{{\left(2\,x\,\sin \left(2\,x\right)-\cos \left(2\,x\right)+1 \right)\,\sin \left(4\,x\right)+\left(\sin \left(2\,x\right)+2\,x\, \cos \left(2\,x\right)\right)\,\cos \left(4\,x\right)-4\,x\,\sin ^2 \left(2\,x\right)-\sin \left(2\,x\right)-4\,x\,\cos ^2\left(2\,x \right)+2\,x\,\cos \left(2\,x\right)}\over{\sin ^2\left(4\,x\right)- 4\,\sin \left(2\,x\right)\,\sin \left(4\,x\right)+\cos ^2\left(4\,x \right)+\left(2-4\,\cos \left(2\,x\right)\right)\,\cos \left(4\,x \right)+4\,\sin ^2\left(2\,x\right)+4\,\cos ^2\left(2\,x\right)-4\, \cos \left(2\,x\right)+1}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.02-0.02
Plot the graph f(x)
The answer [src] 
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Simplify
Numerical answer [src] 
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The graph
Integral of xcosx/sin^3x dx

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

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