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Identical expressions
eight *cos(4x)− two √x+e^(5x+ two)
8 multiply by co sinus of e of (4x)−2√x plus e to the power of (5x plus 2)
eight multiply by co sinus of e of (4x)− two √x plus e to the power of (5x plus two)
8*cos(4x)−2√x+e(5x+2)
8*cos4x−2√x+e5x+2
8cos(4x)−2√x+e^(5x+2)
8cos(4x)−2√x+e(5x+2)
8cos4x−2√x+e5x+2
8cos4x−2√x+e^5x+2
8*cos(4x)−2√x+e^(5x+2)dx
Similar expressions
8*cos(4x)−2√x+e^(5x-2)
8*cos(4x)−2√x-e^(5x+2)

Solving integrals/ 8*cos(4x)−2√x+e^(5x+2)

Integral of 8*cos(4x)−2√x+e^(5x+2) dx

The solution

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  1                                     
  /                                     
 |                                      
 |  /                 ___    5*x + 2\   
 |  \8*cos(4*x) - 2*\/ x  + E       / dx
 |                                      
/                                       
0

$$\int\limits_{0}^{1} \left(e^{5 x + 2} + \left(- 2 \sqrt{x} + 8 \cos{\left(4 x \right)}\right)\right)\, dx$$

Integral(8*cos(4*x) - 2*sqrt(x) + E^(5*x + 2), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = 5 x + 2$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{e^{u}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
    Now substitute $u$ back in:
    
    $\frac{e^{5 x + 2}}{5}$
  Method #2
  1. Rewrite the integrand:
    
    $e^{5 x + 2} = e^{2} e^{5 x}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int e^{2} e^{5 x}\, dx = e^{2} \int e^{5 x}\, dx$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
    So, the result is: $\frac{e^{2} e^{5 x}}{5}$
  Method #3
  1. Rewrite the integrand:
    
    $e^{5 x + 2} = e^{2} e^{5 x}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int e^{2} e^{5 x}\, dx = e^{2} \int e^{5 x}\, dx$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
    So, the result is: $\frac{e^{2} e^{5 x}}{5}$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \sqrt{x}\right)\, dx = - 2 \int \sqrt{x}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$
    So, the result is: $- \frac{4 x^{\frac{3}{2}}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 8 \cos{\left(4 x \right)}\, dx = 8 \int \cos{\left(4 x \right)}\, dx$
    1. Let $u = 4 x$ .
      
      Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{\cos{\left(u \right)}}{4}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{4}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(4 x \right)}}{4}$
    So, the result is: $2 \sin{\left(4 x \right)}$
  The result is: $- \frac{4 x^{\frac{3}{2}}}{3} + 2 \sin{\left(4 x \right)}$
The result is: $- \frac{4 x^{\frac{3}{2}}}{3} + \frac{e^{5 x + 2}}{5} + 2 \sin{\left(4 x \right)}$
Now simplify:

$- \frac{4 x^{\frac{3}{2}}}{3} + \frac{e^{5 x + 2}}{5} + 2 \sin{\left(4 x \right)}$
Add the constant of integration:

$- \frac{4 x^{\frac{3}{2}}}{3} + \frac{e^{5 x + 2}}{5} + 2 \sin{\left(4 x \right)}+ \mathrm{constant}$

The answer is:

$- \frac{4 x^{\frac{3}{2}}}{3} + \frac{e^{5 x + 2}}{5} + 2 \sin{\left(4 x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                         
 |                                                            3/2    5*x + 2
 | /                 ___    5*x + 2\                       4*x      e       
 | \8*cos(4*x) - 2*\/ x  + E       / dx = C + 2*sin(4*x) - ------ + --------
 |                                                           3         5    
/

$$\int \left(e^{5 x + 2} + \left(- 2 \sqrt{x} + 8 \cos{\left(4 x \right)}\right)\right)\, dx = C - \frac{4 x^{\frac{3}{2}}}{3} + \frac{e^{5 x + 2}}{5} + 2 \sin{\left(4 x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                  2    7
  4              e    e 
- - + 2*sin(4) - -- + --
  3              5    5

$$2 \sin{\left(4 \right)} - \frac{e^{2}}{5} - \frac{4}{3} + \frac{e^{7}}{5}$$

                  2    7
  4              e    e 
- - + 2*sin(4) - -- + --
  3              5    5

$$2 \sin{\left(4 \right)} - \frac{e^{2}}{5} - \frac{4}{3} + \frac{e^{7}}{5}$$

-4/3 + 2*sin(4) - exp(2)/5 + exp(7)/5

Numerical answer [src]

215.001882141956

215.001882141956

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

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Integral of 8*cos(4x)−2√x+e^(5x+2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3