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Integral of d{x}:
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Derivative of:
e^x*sin(2*x)
Identical expressions
e^x*sin(two *x)
e to the power of x multiply by sinus of (2 multiply by x)
e to the power of x multiply by sinus of (two multiply by x)
ex*sin(2*x)
ex*sin2*x
e^xsin(2x)
exsin(2x)
exsin2x
e^xsin2x
e^x*sin(2*x)dx

Solving integrals/ e^x*sin(2*x)

Integral of e^xsin(2x) dx

The solution

You have entered [src]

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

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

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

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{x} \sin{\left(2 x \right)}$ :
  
  Let $u{\left(x \right)} = \sin{\left(2 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
  
  Then $\int e^{x} \sin{\left(2 x \right)}\, dx = e^{x} \sin{\left(2 x \right)} - \int 2 e^{x} \cos{\left(2 x \right)}\, dx$ .
2. For the integrand $2 e^{x} \cos{\left(2 x \right)}$ :
  
  Let $u{\left(x \right)} = 2 \cos{\left(2 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
  
  Then $\int e^{x} \sin{\left(2 x \right)}\, dx = e^{x} \sin{\left(2 x \right)} - 2 e^{x} \cos{\left(2 x \right)} + \int \left(- 4 e^{x} \sin{\left(2 x \right)}\right)\, dx$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $5 \int e^{x} \sin{\left(2 x \right)}\, dx = e^{x} \sin{\left(2 x \right)} - 2 e^{x} \cos{\left(2 x \right)}$
  
  Therefore,
  
  $\int e^{x} \sin{\left(2 x \right)}\, dx = \frac{e^{x} \sin{\left(2 x \right)}}{5} - \frac{2 e^{x} \cos{\left(2 x \right)}}{5}$
Now simplify:

$\frac{\left(\sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{x}}{5}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$${{e^{x}\,\left(\sin \left(2\,x\right)-2\,\cos \left(2\,x\right) \right)}\over{5}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2   2*e*cos(2)   e*sin(2)
- - ---------- + --------
5       5           5

$${{e\,\sin 2-2\,e\,\cos 2}\over{5}}+{{2}\over{5}}$$

2   2*e*cos(2)   e*sin(2)
- - ---------- + --------
5       5           5

$$\frac{2}{5} - \frac{2 e \cos{\left(2 \right)}}{5} + \frac{e \sin{\left(2 \right)}}{5}$$

Упростить

Numerical answer [src]

1.34682708790369

1.34682708790369

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of e^xsin(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Integral of e^x*sin(2*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of e^xsin(2x) dx