Integral of xe^xsinx dx

The solution

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

Integral((x*E^x)*sin(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 $\operatorname{dv}{\left(x \right)} = e^{x} \sin{\left(x \right)}$ .

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

$\frac{\left(- \sqrt{2} x \cos{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x \right)}\right) e^{x}}{2}$
Add the constant of integration:

$\frac{\left(- \sqrt{2} x \cos{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x \right)}\right) e^{x}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\left(- \sqrt{2} x \cos{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x \right)}\right) e^{x}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   E*sin(1)
- - + --------
  2      2

$$- \frac{1}{2} + \frac{e \sin{\left(1 \right)}}{2}$$

  1   E*sin(1)
- - + --------
  2      2

$$- \frac{1}{2} + \frac{e \sin{\left(1 \right)}}{2}$$

-1/2 + E*sin(1)/2

Numerical answer [src]

0.643677643589421

0.643677643589421

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of xe^xsinx dx

Limits of integration:

The graph:

Piecewise:

The solution