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Integral of d{x}:
Integral of 1/5x
Integral of (sec(x))^3
Integral of lnx/sqrtx
Integral of x^2*e^(x^3)
Identical expressions
xe^(2x)cos(5x)
xe to the power of (2x) co sinus of e of (5x)
xe(2x)cos(5x)
xe2xcos5x
xe^2xcos5x
xe^(2x)cos(5x)dx

Integral of xe^(2x)cos(5x) dx

The solution

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  1                   
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 |     2*x            
 |  x*E   *cos(5*x) dx
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0

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

Integral((x*E^(2*x))*cos(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 $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x} \cos{\left(5 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. Use integration by parts, noting that the integrand eventually repeats itself.
  1. For the integrand $e^{2 x} \cos{\left(5 x \right)}$ :
    
    Let $u{\left(x \right)} = \cos{\left(5 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
    
    Then $\int e^{2 x} \cos{\left(5 x \right)}\, dx = \frac{e^{2 x} \cos{\left(5 x \right)}}{2} - \int \left(- \frac{5 e^{2 x} \sin{\left(5 x \right)}}{2}\right)\, dx$ .
  2. For the integrand $- \frac{5 e^{2 x} \sin{\left(5 x \right)}}{2}$ :
    
    Let $u{\left(x \right)} = - \frac{5 \sin{\left(5 x \right)}}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
    
    Then $\int e^{2 x} \cos{\left(5 x \right)}\, dx = \frac{5 e^{2 x} \sin{\left(5 x \right)}}{4} + \frac{e^{2 x} \cos{\left(5 x \right)}}{2} + \int \left(- \frac{25 e^{2 x} \cos{\left(5 x \right)}}{4}\right)\, dx$ .
  3. Notice that the integrand has repeated itself, so move it to one side:
    
    $\frac{29 \int e^{2 x} \cos{\left(5 x \right)}\, dx}{4} = \frac{5 e^{2 x} \sin{\left(5 x \right)}}{4} + \frac{e^{2 x} \cos{\left(5 x \right)}}{2}$
    
    Therefore,
    
    $\int e^{2 x} \cos{\left(5 x \right)}\, dx = \frac{5 e^{2 x} \sin{\left(5 x \right)}}{29} + \frac{2 e^{2 x} \cos{\left(5 x \right)}}{29}$
Now evaluate the sub-integral.
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{5 e^{2 x} \sin{\left(5 x \right)}}{29}\, dx = \frac{5 \int e^{2 x} \sin{\left(5 x \right)}\, dx}{29}$
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. For the integrand $e^{2 x} \sin{\left(5 x \right)}$ :
      
      Let $u{\left(x \right)} = \sin{\left(5 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
      
      Then $\int e^{2 x} \sin{\left(5 x \right)}\, dx = \frac{e^{2 x} \sin{\left(5 x \right)}}{2} - \int \frac{5 e^{2 x} \cos{\left(5 x \right)}}{2}\, dx$ .
    2. For the integrand $\frac{5 e^{2 x} \cos{\left(5 x \right)}}{2}$ :
      
      Let $u{\left(x \right)} = \frac{5 \cos{\left(5 x \right)}}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
      
      Then $\int e^{2 x} \sin{\left(5 x \right)}\, dx = \frac{e^{2 x} \sin{\left(5 x \right)}}{2} - \frac{5 e^{2 x} \cos{\left(5 x \right)}}{4} + \int \left(- \frac{25 e^{2 x} \sin{\left(5 x \right)}}{4}\right)\, dx$ .
    3. Notice that the integrand has repeated itself, so move it to one side:
      
      $\frac{29 \int e^{2 x} \sin{\left(5 x \right)}\, dx}{4} = \frac{e^{2 x} \sin{\left(5 x \right)}}{2} - \frac{5 e^{2 x} \cos{\left(5 x \right)}}{4}$
      
      Therefore,
      
      $\int e^{2 x} \sin{\left(5 x \right)}\, dx = \frac{2 e^{2 x} \sin{\left(5 x \right)}}{29} - \frac{5 e^{2 x} \cos{\left(5 x \right)}}{29}$
  So, the result is: $\frac{10 e^{2 x} \sin{\left(5 x \right)}}{841} - \frac{25 e^{2 x} \cos{\left(5 x \right)}}{841}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 e^{2 x} \cos{\left(5 x \right)}}{29}\, dx = \frac{2 \int e^{2 x} \cos{\left(5 x \right)}\, dx}{29}$
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. For the integrand $e^{2 x} \cos{\left(5 x \right)}$ :
      
      Let $u{\left(x \right)} = \cos{\left(5 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
      
      Then $\int e^{2 x} \cos{\left(5 x \right)}\, dx = \frac{e^{2 x} \cos{\left(5 x \right)}}{2} - \int \left(- \frac{5 e^{2 x} \sin{\left(5 x \right)}}{2}\right)\, dx$ .
    2. For the integrand $- \frac{5 e^{2 x} \sin{\left(5 x \right)}}{2}$ :
      
      Let $u{\left(x \right)} = - \frac{5 \sin{\left(5 x \right)}}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
      
      Then $\int e^{2 x} \cos{\left(5 x \right)}\, dx = \frac{5 e^{2 x} \sin{\left(5 x \right)}}{4} + \frac{e^{2 x} \cos{\left(5 x \right)}}{2} + \int \left(- \frac{25 e^{2 x} \cos{\left(5 x \right)}}{4}\right)\, dx$ .
    3. Notice that the integrand has repeated itself, so move it to one side:
      
      $\frac{29 \int e^{2 x} \cos{\left(5 x \right)}\, dx}{4} = \frac{5 e^{2 x} \sin{\left(5 x \right)}}{4} + \frac{e^{2 x} \cos{\left(5 x \right)}}{2}$
      
      Therefore,
      
      $\int e^{2 x} \cos{\left(5 x \right)}\, dx = \frac{5 e^{2 x} \sin{\left(5 x \right)}}{29} + \frac{2 e^{2 x} \cos{\left(5 x \right)}}{29}$
  So, the result is: $\frac{10 e^{2 x} \sin{\left(5 x \right)}}{841} + \frac{4 e^{2 x} \cos{\left(5 x \right)}}{841}$
The result is: $\frac{20 e^{2 x} \sin{\left(5 x \right)}}{841} - \frac{21 e^{2 x} \cos{\left(5 x \right)}}{841}$
Now simplify:

$\frac{\left(145 x \sin{\left(5 x \right)} + 58 x \cos{\left(5 x \right)} - 20 \sin{\left(5 x \right)} + 21 \cos{\left(5 x \right)}\right) e^{2 x}}{841}$
Add the constant of integration:

$\frac{\left(145 x \sin{\left(5 x \right)} + 58 x \cos{\left(5 x \right)} - 20 \sin{\left(5 x \right)} + 21 \cos{\left(5 x \right)}\right) e^{2 x}}{841}+ \mathrm{constant}$

The answer is:

$\frac{\left(145 x \sin{\left(5 x \right)} + 58 x \cos{\left(5 x \right)} - 20 \sin{\left(5 x \right)} + 21 \cos{\left(5 x \right)}\right) e^{2 x}}{841}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                    
 |                            /            2*x      2*x         \       2*x                         2*x
 |    2*x                     |2*cos(5*x)*e      5*e   *sin(5*x)|   20*e   *sin(5*x)   21*cos(5*x)*e   
 | x*E   *cos(5*x) dx = C + x*|--------------- + ---------------| - ---------------- + ----------------
 |                            \       29                29      /         841                841       
/

$$\int e^{2 x} x \cos{\left(5 x \right)}\, dx = C + x \left(\frac{5 e^{2 x} \sin{\left(5 x \right)}}{29} + \frac{2 e^{2 x} \cos{\left(5 x \right)}}{29}\right) - \frac{20 e^{2 x} \sin{\left(5 x \right)}}{841} + \frac{21 e^{2 x} \cos{\left(5 x \right)}}{841}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                   2        2       
   21   79*cos(5)*e    125*e *sin(5)
- --- + ------------ + -------------
  841       841             841

$$\frac{125 e^{2} \sin{\left(5 \right)}}{841} - \frac{21}{841} + \frac{79 e^{2} \cos{\left(5 \right)}}{841}$$

                   2        2       
   21   79*cos(5)*e    125*e *sin(5)
- --- + ------------ + -------------
  841       841             841

$$\frac{125 e^{2} \sin{\left(5 \right)}}{841} - \frac{21}{841} + \frac{79 e^{2} \cos{\left(5 \right)}}{841}$$

-21/841 + 79*cos(5)*exp(2)/841 + 125*exp(2)*sin(5)/841

Numerical answer [src]

-0.881224125080771

-0.881224125080771

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

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Identical expressions

Integral of xe^(2x)cos(5x) dx

Limits of integration:

The graph:

Piecewise:

The solution