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Identical expressions
((eight ^x)-5sin4x+ seven)
((8 to the power of x) minus 5 sinus of 4x plus 7)
((eight to the power of x) minus 5 sinus of 4x plus seven)
((8x)-5sin4x+7)
8x-5sin4x+7
8^x-5sin4x+7
((8^x)-5sin4x+7)dx
Similar expressions
((8^x)+5sin4x+7)
((8^x)-5sin4x-7)

Solving integrals/ ((8^x)-5sin4x+7)

Integral of ((8^x)-5sin4x+7) dx

The solution

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  1                         
  /                         
 |                          
 |  / x                 \   
 |  \8  - 5*sin(4*x) + 7/ dx
 |                          
/                           
0

$$\int\limits_{0}^{1} \left(8^{x} - 5 \sin{\left(4 x \right)} + 7\right)\, dx$$

Integral(8^x - 5*sin(4*x) + 7, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of an exponential function is itself divided by the natural logarithm of the base.
  
  $\int 8^{x}\, dx = \frac{8^{x}}{\log{\left(8 \right)}}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 5 \sin{\left(4 x \right)}\right)\, dx = - \int 5 \sin{\left(4 x \right)}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 5 \sin{\left(4 x \right)}\, dx = 5 \int \sin{\left(4 x \right)}\, dx$
    1. Let $u = 4 x$ .
      
      Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{\sin{\left(u \right)}}{16}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{\sin{\left(u \right)}}{4}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
        
        The integral of sine is negative cosine:
        
        $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
        
        So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(4 x \right)}}{4}$
    So, the result is: $- \frac{5 \cos{\left(4 x \right)}}{4}$
  So, the result is: $\frac{5 \cos{\left(4 x \right)}}{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 7\, dx = 7 x$
The result is: $\frac{8^{x}}{\log{\left(8 \right)}} + 7 x + \frac{5 \cos{\left(4 x \right)}}{4}$
Now simplify:

$\frac{2^{3 x}}{\log{\left(8 \right)}} + 7 x + \frac{5 \cos{\left(4 x \right)}}{4}$
Add the constant of integration:

$\frac{2^{3 x}}{\log{\left(8 \right)}} + 7 x + \frac{5 \cos{\left(4 x \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{2^{3 x}}{\log{\left(8 \right)}} + 7 x + \frac{5 \cos{\left(4 x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                        
 |                                                      x  
 | / x                 \                5*cos(4*x)     8   
 | \8  - 5*sin(4*x) + 7/ dx = C + 7*x + ---------- + ------
 |                                          4        log(8)
/

$${{5\,\cos \left(4\,x\right)}\over{4}}+{{8^{x}}\over{\log 8}}+7\,x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

23   5*cos(4)      7    
-- + -------- + --------
4       4       3*log(2)

$${{\left(5\,\cos 4+23\right)\,\log 8+28}\over{4\,\log 8}}$$

23   5*cos(4)      7    
-- + -------- + --------
4       4       3*log(2)

$$\frac{5 \cos{\left(4 \right)}}{4} + \frac{7}{3 \log{\left(2 \right)}} + \frac{23}{4}$$

Упростить

Numerical answer [src]

8.2992339026614

8.2992339026614

The graph

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

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Integral of ((8^x)-5sin4x+7) dx

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The solution