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Integral of xcosx/sin^3x
Identical expressions
xcosx/sin^3x
x co sinus of e of x divide by sinus of cubed x
xcosx/sin3x
xcosx/sin³x
xcosx/sin to the power of 3x
xcosx divide by sin^3x
xcosx/sin^3xdx

Solving integrals/ xcosx/sin^3x

Integral of xcosx/sin^3x dx

The solution

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

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

Simplify

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

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

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

To find $v{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
  
  Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
  
  $\int \frac{1}{u^{3}}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$
  Now substitute $u$ back in:
  
  $- \frac{1}{2 \sin^{2}{\left(x \right)}}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{1}{2 \sin^{2}{\left(x \right)}}\right)\, dx = - \frac{\int \frac{1}{\sin^{2}{\left(x \right)}}\, dx}{2}$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
So, the result is: $\frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)}}$
Now simplify:

$- \frac{x}{2 \sin^{2}{\left(x \right)}} - \frac{1}{2 \tan{\left(x \right)}}$
Add the constant of integration:

$- \frac{x}{2 \sin^{2}{\left(x \right)}} - \frac{1}{2 \tan{\left(x \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{x}{2 \sin^{2}{\left(x \right)}} - \frac{1}{2 \tan{\left(x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$${{\left(2\,x\,\sin \left(2\,x\right)-\cos \left(2\,x\right)+1 \right)\,\sin \left(4\,x\right)+\left(\sin \left(2\,x\right)+2\,x\, \cos \left(2\,x\right)\right)\,\cos \left(4\,x\right)-4\,x\,\sin ^2 \left(2\,x\right)-\sin \left(2\,x\right)-4\,x\,\cos ^2\left(2\,x \right)+2\,x\,\cos \left(2\,x\right)}\over{\sin ^2\left(4\,x\right)- 4\,\sin \left(2\,x\right)\,\sin \left(4\,x\right)+\cos ^2\left(4\,x \right)+\left(2-4\,\cos \left(2\,x\right)\right)\,\cos \left(4\,x \right)+4\,\sin ^2\left(2\,x\right)+4\,\cos ^2\left(2\,x\right)-4\, \cos \left(2\,x\right)+1}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Simplify

Numerical answer [src]

0.0

0.0

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of xcosx/sin^3x dx

Limits of integration:

The graph:

Piecewise:

The solution