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Identical expressions
x^ two *sin5xdx
x squared multiply by sinus of 5xdx
x to the power of two multiply by sinus of 5xdx
x2*sin5xdx
x²*sin5xdx
x to the power of 2*sin5xdx
x^2sin5xdx
x2sin5xdx

Integral of x^2*sin5xdx dx

The solution

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  1               
  /               
 |                
 |   2            
 |  x *sin(5*x) dx
 |                
/                 
0

$$\int\limits_{0}^{1} x^{2} \sin{\left(5 x \right)}\, dx$$

Integral(x^2*sin(5*x), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(5 x \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\sin{\left(u \right)}}{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(5 x \right)}}{5}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = - \frac{2 x}{5}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(5 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - \frac{2}{5}$ .

To find $v{\left(x \right)}$ :
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\cos{\left(u \right)}}{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(5 x \right)}}{5}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{2 \sin{\left(5 x \right)}}{25}\right)\, dx = - \frac{2 \int \sin{\left(5 x \right)}\, dx}{25}$
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\sin{\left(u \right)}}{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(5 x \right)}}{5}$
So, the result is: $\frac{2 \cos{\left(5 x \right)}}{125}$
Add the constant of integration:

$- \frac{x^{2} \cos{\left(5 x \right)}}{5} + \frac{2 x \sin{\left(5 x \right)}}{25} + \frac{2 \cos{\left(5 x \right)}}{125}+ \mathrm{constant}$

The answer is:

$- \frac{x^{2} \cos{\left(5 x \right)}}{5} + \frac{2 x \sin{\left(5 x \right)}}{25} + \frac{2 \cos{\left(5 x \right)}}{125}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                    2                        
 |  2                   2*cos(5*x)   x *cos(5*x)   2*x*sin(5*x)
 | x *sin(5*x) dx = C + ---------- - ----------- + ------------
 |                         125            5             25     
/

$$\int x^{2} \sin{\left(5 x \right)}\, dx = C - \frac{x^{2} \cos{\left(5 x \right)}}{5} + \frac{2 x \sin{\left(5 x \right)}}{25} + \frac{2 \cos{\left(5 x \right)}}{125}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   2    23*cos(5)   2*sin(5)
- --- - --------- + --------
  125      125         25

$$\frac{2 \sin{\left(5 \right)}}{25} - \frac{23 \cos{\left(5 \right)}}{125} - \frac{2}{125}$$

   2    23*cos(5)   2*sin(5)
- --- - --------- + --------
  125      125         25

$$\frac{2 \sin{\left(5 \right)}}{25} - \frac{23 \cos{\left(5 \right)}}{125} - \frac{2}{125}$$

-2/125 - 23*cos(5)/125 + 2*sin(5)/25

Numerical answer [src]

-0.144907784098285

-0.144907784098285

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

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Identical expressions

Integral of x^2*sin5xdx dx

Limits of integration:

The graph:

Piecewise:

The solution