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Solving integrals/ sinx+cos3x-2^x

Integral of sinx+cos3x-2^x dx

The solution

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  1                            
  /                            
 |                             
 |  /                     x\   
 |  \sin(x) + cos(3*x) - 2 / dx
 |                             
/                              
0

$$\int\limits_{0}^{1} \left(- 2^{x} + \left(\sin{\left(x \right)} + \cos{\left(3 x \right)}\right)\right)\, dx$$

Integral(sin(x) + cos(3*x) - 2^x, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 2^{x}\right)\, dx = - \int 2^{x}\, dx$
  1. The integral of an exponential function is itself divided by the natural logarithm of the base.
    
    $\int 2^{x}\, dx = \frac{2^{x}}{\log{\left(2 \right)}}$
  So, the result is: $- \frac{2^{x}}{\log{\left(2 \right)}}$
1. Integrate term-by-term:
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  1. Let $u = 3 x$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{\cos{\left(u \right)}}{3}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(3 x \right)}}{3}$
  The result is: $\frac{\sin{\left(3 x \right)}}{3} - \cos{\left(x \right)}$
The result is: $- \frac{2^{x}}{\log{\left(2 \right)}} + \frac{\sin{\left(3 x \right)}}{3} - \cos{\left(x \right)}$
Add the constant of integration:

$- \frac{2^{x}}{\log{\left(2 \right)}} + \frac{\sin{\left(3 x \right)}}{3} - \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$- \frac{2^{x}}{\log{\left(2 \right)}} + \frac{\sin{\left(3 x \right)}}{3} - \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

      1               sin(3)
1 - ------ - cos(1) + ------
    log(2)              3

$$- \frac{1}{\log{\left(2 \right)}} - \cos{\left(1 \right)} + \frac{\sin{\left(3 \right)}}{3} + 1$$

      1               sin(3)
1 - ------ - cos(1) + ------
    log(2)              3

$$- \frac{1}{\log{\left(2 \right)}} - \cos{\left(1 \right)} + \frac{\sin{\left(3 \right)}}{3} + 1$$

1 - 1/log(2) - cos(1) + sin(3)/3

Numerical answer [src]

-0.935957344070481

-0.935957344070481

The graph

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

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Integral of sinx+cos3x-2^x dx

Limits of integration:

The graph:

Piecewise:

The solution