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xsinx/cos^3xdx

Solving integrals/ xsinx/cos^3x

Integral of xsinx/cos^3x dx

The solution

You have entered [src]

 pi            
 --            
 4             
  /            
 |             
 |  x*sin(x)   
 |  -------- dx
 |     3       
 |  cos (x)    
 |             
/              
0

$$\int\limits_{0}^{\frac{\pi}{4}} \frac{x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}\, dx$$

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

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                                                   3                                                                                      2                                           4            
                /       ___\                           /       ___\                                                                           /       ___\                                /       ___\             
              2*\-1 + \/ 2 /                         2*\-1 + \/ 2 /                                   pi                                   pi*\-1 + \/ 2 /                             pi*\-1 + \/ 2 /             
- ------------------------------------- + ------------------------------------- + ----------------------------------------- + ----------------------------------------- + -----------------------------------------
                    2                 4                     2                 4     /                  2                 4\     /                  2                 4\     /                  2                 4\
        /       ___\      /       ___\          /       ___\      /       ___\      |      /       ___\      /       ___\ |     |      /       ___\      /       ___\ |     |      /       ___\      /       ___\ |
  2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 /    2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 /    4*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /   2*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /   4*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /

$$- \frac{2 \left(-1 + \sqrt{2}\right)}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{4}}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{2 \left(-1 + \sqrt{2}\right)^{3}}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{2}}{2 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{\pi}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)}$$

                                                                   3                                                                                      2                                           4            
                /       ___\                           /       ___\                                                                           /       ___\                                /       ___\             
              2*\-1 + \/ 2 /                         2*\-1 + \/ 2 /                                   pi                                   pi*\-1 + \/ 2 /                             pi*\-1 + \/ 2 /             
- ------------------------------------- + ------------------------------------- + ----------------------------------------- + ----------------------------------------- + -----------------------------------------
                    2                 4                     2                 4     /                  2                 4\     /                  2                 4\     /                  2                 4\
        /       ___\      /       ___\          /       ___\      /       ___\      |      /       ___\      /       ___\ |     |      /       ___\      /       ___\ |     |      /       ___\      /       ___\ |
  2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 /    2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 /    4*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /   2*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /   4*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /

Simplify

Numerical answer [src]

0.285398163397448

0.285398163397448

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of xsinx/cos^3x dx

Limits of integration:

The graph:

Piecewise:

The solution