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Similar expressions
x^2*sqrt(2)cosx

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

The solution

You have entered [src]

 pi                   
 --                   
 4                    
  /                   
 |                    
 |   2   ___          
 |  x *\/ 2 *cos(x) dx
 |                    
/                     
0

$$\int\limits_{0}^{\frac{\pi}{4}} \sqrt{2} x^{2} \cos{\left(x \right)}\, dx$$

Integral((x^2*sqrt(2))*cos(x), (x, 0, pi/4))

Detail solution

Use integration by parts:

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

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

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

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

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

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:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

      /               ___     ___   2\
  ___ |    ___   pi*\/ 2    \/ 2 *pi |
\/ 2 *|- \/ 2  + -------- + ---------|
      \             4           32   /

$$\sqrt{2} \left(- \sqrt{2} + \frac{\sqrt{2} \pi^{2}}{32} + \frac{\sqrt{2} \pi}{4}\right)$$

      /               ___     ___   2\
  ___ |    ___   pi*\/ 2    \/ 2 *pi |
\/ 2 *|- \/ 2  + -------- + ---------|
      \             4           32   /

$$\sqrt{2} \left(- \sqrt{2} + \frac{\sqrt{2} \pi^{2}}{32} + \frac{\sqrt{2} \pi}{4}\right)$$

sqrt(2)*(-sqrt(2) + pi*sqrt(2)/4 + sqrt(2)*pi^2/32)

Numerical answer [src]

0.187646601862982

0.187646601862982

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution