Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of sin^6xcos^4x
Integral of xcosx/sin^2x
Integral of e^(−3x)cos2x
Integral of e^x(1-cotx+cosec^2x)dx
Identical expressions
xcosx/sin^2x
x co sinus of e of x divide by sinus of squared x
xcosx/sin2x
xcosx/sin²x
xcosx/sin to the power of 2x
xcosx divide by sin^2x
xcosx/sin^2xdx

Solving integrals/ xcosx/sin^2x

Integral of xcosx/sin^2x dx

The solution

You have entered [src]

  1            
  /            
 |             
 |  x*cos(x)   
 |  -------- dx
 |     2       
 |  sin (x)    
 |             
/              
0

$$\int\limits_{0}^{1} \frac{x \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx$$

Integral(x*cos(x)/(sin(x)^2), (x, 0, 1))

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 $\operatorname{dv}{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$ .

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

$\int \left(- \frac{1}{\sin{\left(x \right)}}\right)\, dx = - \int \frac{1}{\sin{\left(x \right)}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2}$
So, the result is: $- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                             
 |                                                              
 | x*cos(x)          log(-1 + cos(x))   log(1 + cos(x))     x   
 | -------- dx = C + ---------------- - --------------- - ------
 |    2                     2                  2          sin(x)
 | sin (x)                                                      
 |                                                              
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Simplify

Numerical answer [src]

43.9906157628331

43.9906157628331

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^2x dx

Limits of integration:

The graph:

Piecewise:

The solution