Mister Exam

Other calculators

Solving integrals/ 2^(x*(-3))*sin(2*x)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 oo                    
  /                    
 |                     
 |   x*(-3)            
 |  2      *sin(2*x) dx
 |                     
/                      
0
∫0∞2(−3)xsin⁡(2x) dx\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

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

    ∫2⋅2(−3)xsin⁡(x)cos⁡(x) dx=2∫2(−3)xsin⁡(x)cos⁡(x) dx\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

      sin⁡2(x)4⋅23x+9⋅23xlog⁡(2)2−3log⁡(2)sin⁡(x)cos⁡(x)4⋅23x+9⋅23xlog⁡(2)2−cos⁡2(x)4⋅23x+9⋅23xlog⁡(2)2\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: 2sin⁡2(x)4⋅23x+9⋅23xlog⁡(2)2−6log⁡(2)sin⁡(x)cos⁡(x)4⋅23x+9⋅23xlog⁡(2)2−2cos⁡2(x)4⋅23x+9⋅23xlog⁡(2)2\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}}

  2. Now simplify:

    −8−x(log⁡(8)sin⁡(2x)+2cos⁡(2x))4+9log⁡(2)2- \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}}

  3. Add the constant of integration:

    −8−x(log⁡(8)sin⁡(2x)+2cos⁡(2x))4+9log⁡(2)2+constant- \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:

−8−x(log⁡(8)sin⁡(2x)+2cos⁡(2x))4+9log⁡(2)2+constant- \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)
∫2(−3)xsin⁡(2x) dx=C+2sin⁡2(x)4⋅23x+9⋅23xlog⁡(2)2−6log⁡(2)sin⁡(x)cos⁡(x)4⋅23x+9⋅23xlog⁡(2)2−2cos⁡2(x)4⋅23x+9⋅23xlog⁡(2)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
0.001.000.100.200.300.400.500.600.700.800.900.25-0.25
Plot the graph f(x)
The answer [src] 
      2      
-------------
         2   
4 + 9*log (2)
24+9log⁡(2)2\frac{2}{4 + 9 \log{\left(2 \right)}^{2}} 
=
= 
      2      
-------------
         2   
4 + 9*log (2)
24+9log⁡(2)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.

    © Mister Exam – Calculator