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Integral of d{x}:
Integral of 1/(sin^4x*cos^4x)
Integral of 8cos4x-2sqrt(x)+e^(5x+2)
Integral of ctg(2x-1)
Integral of xdx^2
Identical expressions
8cos4x- two sqrt(x)+e^(5x+2)
8 co sinus of e of 4x minus 2 square root of (x) plus e to the power of (5x plus 2)
8 co sinus of e of 4x minus two square root of (x) plus e to the power of (5x plus 2)
8cos4x-2√(x)+e^(5x+2)
8cos4x-2sqrt(x)+e(5x+2)
8cos4x-2sqrtx+e5x+2
8cos4x-2sqrtx+e^5x+2
8cos4x-2sqrt(x)+e^(5x+2)dx
Similar expressions
8cos4x-2sqrt(x)+e^(5x-2)
8cos4x-2sqrt(x)-e^(5x+2)
8cos4x+2sqrt(x)+e^(5x+2)

Solving integrals/ 8cos4x-2sqrt(x)+e^(5x+2)

Integral of 8cos4x-2sqrt(x)+e^(5x+2) dx

The solution

You have entered [src]

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

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

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

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 2 \sqrt{x}\right)\, dx = - \int 2 \sqrt{x}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \sqrt{x}\, dx = 2 \int \sqrt{x}\, dx$
    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: $\frac{4 x^{\frac{3}{2}}}{3}$
  So, the result is: $- \frac{4 x^{\frac{3}{2}}}{3}$
1. There are multiple ways to do this integral.
  Method #1
  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. There are multiple ways to do this integral.
      
      Method #1
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      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:
      
      $\frac{e^{5 x}}{5}$
      
      Method #2
      
      Let $u = e^{5 x}$ .
      
      Then let $du = 5 e^{5 x} dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{1}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{5}\, du = \frac{\int 1\, du}{5}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
    So, the result is: $\frac{e^{2} e^{5 x}}{5}$
  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 $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      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. 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 $\frac{du}{4}$ :
    
    $\int \frac{\cos{\left(u \right)}}{16}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{4}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{4}$
    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: $- \frac{4 x^{\frac{3}{2}}}{3} + \frac{e^{2} e^{5 x}}{5} + 2 \sin{\left(4 x \right)}$
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    2  5*x
 | /                 ___    5*x + 2\                       4*x      e *e   
 | \8*cos(4*x) - 2*\/ x  + e       / dx = C + 2*sin(4*x) - ------ + -------
 |                                                           3         5   
/

$$2\,\sin \left(4\,x\right)+{{e^{5\,x+2}}\over{5}}-{{4\,x^{{{3}\over{ 2}}}}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$${{30\,\sin 4+3\,e^7-3\,e^2-20}\over{15}}$$

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

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

Simplify

Numerical answer [src]

215.001882141956

215.001882141956

The graph

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

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Integral of 8cos4x-2sqrt(x)+e^(5x+2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2