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Integral of e^2xsinx dx

The solution

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  1               
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 |  E *x*sin(x) dx
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0

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

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

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

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:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                    2
(-cos(1) + sin(1))*e

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

                    2
(-cos(1) + sin(1))*e

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

(-cos(1) + sin(1))*exp(2)

Numerical answer [src]

2.2253522639267

2.2253522639267

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

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Identical expressions

Similar expressions

Integral of e^2xsinx dx

Limits of integration:

The graph:

Piecewise:

The solution