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Solving integrals/ -x^2*cos(x)

Integral of -x^2*cos(x) dx

The solution

You have entered [src]

 3*pi             
 ----             
  2               
   /              
  |               
  |    2          
  |  -x *cos(x) dx
  |               
 /                
 pi

$$\int\limits_{\pi}^{\frac{3 \pi}{2}} - x^{2} \cos{\left(x \right)}\, dx$$

Integral((-x^2)*cos(x), (x, pi, 3*pi/2))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = - x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - 2 x$ .

To find $v{\left(x \right)}$ :
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = - 2 x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = -2$ .

To find $v{\left(x \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int 2 \cos{\left(x \right)}\, dx = 2 \int \cos{\left(x \right)}\, dx$
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
So, the result is: $2 \sin{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int - x^{2} \cos{\left(x \right)}\, dx = C - x^{2} \sin{\left(x \right)} - 2 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                2
            9*pi 
-2 - 2*pi + -----
              4

$$- 2 \pi - 2 + \frac{9 \pi^{2}}{4}$$

                2
            9*pi 
-2 - 2*pi + -----
              4

$$- 2 \pi - 2 + \frac{9 \pi^{2}}{4}$$

Упростить

Numerical answer [src]

13.9234245952715

13.9234245952715

The graph

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

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Identical expressions

Similar expressions

Integral of -x^2*cos(x) dx

Limits of integration:

The graph:

Piecewise:

The solution