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Identical expressions
two ^(x*(- three))*sin(two *x)
2 to the power of (x multiply by ( minus 3)) multiply by sinus of (2 multiply by x)
two to the power of (x multiply by ( minus three)) multiply by sinus of (two multiply by x)
2(x*(-3))*sin(2*x)
2x*-3*sin2*x
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2(x(-3))sin(2x)
2x-3sin2x
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2^(x*(-3))*sin(2*x)dx
Similar expressions
2^(x*(3))*sin(2*x)

Integral of 2^(x(-3))sin(2*x) dx

The solution

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 oo                    
  /                    
 |                     
 |   x*(-3)            
 |  2      *sin(2*x) dx
 |                     
/                      
0

$$\int\limits_{0}^{\infty} 2^{\left(-3\right) x} \sin{\left(2 x \right)}\, dx$$

Integral(2^(x*(-3))*sin(2*x), (x, 0, oo))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 2 \cdot 2^{\left(-3\right) x} \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int 2^{\left(-3\right) x} \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{\sin^{2}{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}} - \frac{3 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}} - \frac{\cos^{2}{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}}$
So, the result is: $\frac{2 \sin^{2}{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}} - \frac{6 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}} - \frac{2 \cos^{2}{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}}$
Now simplify:

$- \frac{8^{- x} \left(\log{\left(8 \right)} \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right)}{4 + 9 \log{\left(2 \right)}^{2}}$
Add the constant of integration:

$- \frac{8^{- x} \left(\log{\left(8 \right)} \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right)}{4 + 9 \log{\left(2 \right)}^{2}}+ \mathrm{constant}$

The answer is:

$- \frac{8^{- x} \left(\log{\left(8 \right)} \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right)}{4 + 9 \log{\left(2 \right)}^{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                     
 |                                       2                         2                                    
 |  x*(-3)                          2*cos (x)                 2*sin (x)           6*cos(x)*log(2)*sin(x)
 | 2      *sin(2*x) dx = C - ----------------------- + ----------------------- - -----------------------
 |                              3*x      3*x    2         3*x      3*x    2         3*x      3*x    2   
/                            4*2    + 9*2   *log (2)   4*2    + 9*2   *log (2)   4*2    + 9*2   *log (2)

$$\int 2^{\left(-3\right) x} \sin{\left(2 x \right)}\, dx = C + \frac{2 \sin^{2}{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}} - \frac{6 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}} - \frac{2 \cos^{2}{\left(x \right)}}{4 \cdot 2^{3 x} + 9 \cdot 2^{3 x} \log{\left(2 \right)}^{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      2      
-------------
         2   
4 + 9*log (2)

$$\frac{2}{4 + 9 \log{\left(2 \right)}^{2}}$$

      2      
-------------
         2   
4 + 9*log (2)

$$\frac{2}{4 + 9 \log{\left(2 \right)}^{2}}$$

2/(4 + 9*log(2)^2)

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

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Identical expressions

Similar expressions

Integral of 2^(x(-3))sin(2*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 2^(x*(-3))*sin(2*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of 2^(x(-3))sin(2*x) dx