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Identical expressions
x^ two *cos7x
x squared multiply by co sinus of e of 7x
x to the power of two multiply by co sinus of e of 7x
x2*cos7x
x²*cos7x
x to the power of 2*cos7x
x^2cos7x
x2cos7x
x^2*cos7xdx

Integral of x^2*cos7x dx

The solution

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 157              
 ---              
 100              
  /               
 |                
 |   2            
 |  x *cos(7*x) dx
 |                
/                 
0

$$\int\limits_{0}^{\frac{157}{100}} x^{2} \cos{\left(7 x \right)}\, dx$$

Integral(x^2*cos(7*x), (x, 0, 157/100))

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 $\operatorname{dv}{\left(x \right)} = \cos{\left(7 x \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 7 x$ .
  
  Then let $du = 7 dx$ and substitute $\frac{du}{7}$ :
  
  $\int \frac{\cos{\left(u \right)}}{7}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{7}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{7}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(7 x \right)}}{7}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = \frac{2 x}{7}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(7 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = \frac{2}{7}$ .

To find $v{\left(x \right)}$ :
1. Let $u = 7 x$ .
  
  Then let $du = 7 dx$ and substitute $\frac{du}{7}$ :
  
  $\int \frac{\sin{\left(u \right)}}{7}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{7}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{7}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(7 x \right)}}{7}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{2 \cos{\left(7 x \right)}}{49}\right)\, dx = - \frac{2 \int \cos{\left(7 x \right)}\, dx}{49}$
1. Let $u = 7 x$ .
  
  Then let $du = 7 dx$ and substitute $\frac{du}{7}$ :
  
  $\int \frac{\cos{\left(u \right)}}{7}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{7}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{7}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(7 x \right)}}{7}$
So, the result is: $- \frac{2 \sin{\left(7 x \right)}}{343}$
Add the constant of integration:

$\frac{x^{2} \sin{\left(7 x \right)}}{7} + \frac{2 x \cos{\left(7 x \right)}}{49} - \frac{2 \sin{\left(7 x \right)}}{343}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \sin{\left(7 x \right)}}{7} + \frac{2 x \cos{\left(7 x \right)}}{49} - \frac{2 \sin{\left(7 x \right)}}{343}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                    2                        
 |  2                   2*sin(7*x)   x *sin(7*x)   2*x*cos(7*x)
 | x *cos(7*x) dx = C - ---------- + ----------- + ------------
 |                         343            7             49     
/

$$\int x^{2} \cos{\left(7 x \right)}\, dx = C + \frac{x^{2} \sin{\left(7 x \right)}}{7} + \frac{2 x \cos{\left(7 x \right)}}{49} - \frac{2 \sin{\left(7 x \right)}}{343}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       /1099\              /1099\
157*cos|----|   1187801*sin|----|
       \100 /              \100 /
------------- + -----------------
     2450            3430000

$$\frac{1187801 \sin{\left(\frac{1099}{100} \right)}}{3430000} + \frac{157 \cos{\left(\frac{1099}{100} \right)}}{2450}$$

       /1099\              /1099\
157*cos|----|   1187801*sin|----|
       \100 /              \100 /
------------- + -----------------
     2450            3430000

$$\frac{1187801 \sin{\left(\frac{1099}{100} \right)}}{3430000} + \frac{157 \cos{\left(\frac{1099}{100} \right)}}{2450}$$

157*cos(1099/100)/2450 + 1187801*sin(1099/100)/3430000

Numerical answer [src]

-0.34664949505188

-0.34664949505188

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

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Identical expressions

Integral of x^2*cos7x dx

Limits of integration:

The graph:

Piecewise:

The solution