Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x/sin(x)
Integral of atan(x)
Integral of cosnx
Integral of 1/(sqrt(1-x^2))
Identical expressions
(e^x-e^-x)/2x
(e to the power of x minus e to the power of minus x) divide by 2x
(ex-e-x)/2x
ex-e-x/2x
e^x-e^-x/2x
(e^x-e^-x) divide by 2x
(e^x-e^-x)/2xdx
Similar expressions
(e^x+e^-x)/2x
(e^x-e^+x)/2x

Solving integrals/ (e^x-e^-x)/2x

Integral of (e^x-e^-x)/2x dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |  / x    -x\     
 |  \e  - e  /*x   
 |  ------------ dx
 |       2         
 |                 
/                  
3

$$\int\limits_{3}^{1} \frac{x \left(e^{x} - e^{- x}\right)}{2}\, dx$$

Integral((E^x - 1/E^x)*x/2, (x, 3, 1))

Detail solution

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

$\int \frac{x \left(e^{x} - e^{- x}\right)}{2}\, dx = \frac{\int x \left(e^{x} - e^{- x}\right)\, dx}{2}$
1. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $x \left(e^{x} - e^{- x}\right) = \left(x e^{2 x} - x\right) e^{- x}$
  2. Let $u = - x$ .
    
    Then let $du = - dx$ and substitute $du$ :
    
    $\int \left(- u e^{2 u} + u\right) e^{- u}\, du$
    1. Rewrite the integrand:
      
      $\left(- u e^{2 u} + u\right) e^{- u} = - u e^{u} + u e^{- u}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u e^{u}\right)\, du = - \int u e^{u}\, du$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now evaluate the sub-integral.
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $- u e^{u} + e^{u}$
      
      Let $u = - u$ .
      
      Then let $du = - du$ and substitute $du$ :
      
      $\int u e^{u}\, du$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now evaluate the sub-integral.
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now substitute $u$ back in:
      
      $- u e^{- u} - e^{- u}$
      
      The result is: $- u e^{u} - u e^{- u} + e^{u} - e^{- u}$
    Now substitute $u$ back in:
    
    $x e^{x} + x e^{- x} - e^{x} + e^{- x}$
  Method #2
  1. Rewrite the integrand:
    
    $x \left(e^{x} - e^{- x}\right) = x e^{x} - x e^{- x}$
  2. Integrate term-by-term:
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of the exponential function is itself.
      
      $\int e^{x}\, dx = e^{x}$
      
      Now evaluate the sub-integral.
    2. The integral of the exponential function is itself.
      
      $\int e^{x}\, dx = e^{x}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- x e^{- x}\right)\, dx = - \int x e^{- x}\, dx$
      
      Let $u = - x$ .
      
      Then let $du = - dx$ and substitute $du$ :
      
      $\int u e^{u}\, du$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now evaluate the sub-integral.
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now substitute $u$ back in:
      
      $- x e^{- x} - e^{- x}$
      
      So, the result is: $x e^{- x} + e^{- x}$
    The result is: $x e^{x} + x e^{- x} - e^{x} + e^{- x}$
So, the result is: $\frac{x e^{x}}{2} + \frac{x e^{- x}}{2} - \frac{e^{x}}{2} + \frac{e^{- x}}{2}$
Now simplify:

$\frac{\left(x + \left(x - 1\right) e^{2 x} + 1\right) e^{- x}}{2}$
Add the constant of integration:

$\frac{\left(x + \left(x - 1\right) e^{2 x} + 1\right) e^{- x}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\left(x + \left(x - 1\right) e^{2 x} + 1\right) e^{- x}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                              
 | / x    -x\             -x    x      x      -x
 | \e  - e  /*x          e     e    x*e    x*e  
 | ------------ dx = C + --- - -- + ---- + -----
 |      2                 2    2     2       2  
 |                                              
/

$${{\left(x-1\right)\,e^{x}-\left(-x-1\right)\,e^ {- x }}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   3      -3    -1
- e  - 2*e   + e

$$-{{e^ {- 3 }\,\left(2\,e^6+4\right)-2\,e^ {- 1 }}\over{2}}$$

   3      -3    -1
- e  - 2*e   + e

$$- e^{3} - \frac{2}{e^{3}} + e^{-1}$$

Simplify

Numerical answer [src]

-19.817231618752

-19.817231618752

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (e^x-e^-x)/2x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2