Mister Exam

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Identical expressions
(e^sin5x)*cos5x
(e to the power of sinus of 5x) multiply by co sinus of e of 5x
(esin5x)*cos5x
esin5x*cos5x
(e^sin5x)cos5x
(esin5x)cos5x
esin5xcos5x
e^sin5xcos5x
(e^sin5x)*cos5xdx

Integral of (e^sin5x)*cos5x dx

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |   sin(5*x)            
 |  E        *cos(5*x) dx
 |                       
/                        
0

$$\int\limits_{0}^{1} e^{\sin{\left(5 x \right)}} \cos{\left(5 x \right)}\, dx$$

Integral(E^sin(5*x)*cos(5*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sin{\left(5 x \right)}$ .
  
  Then let $du = 5 \cos{\left(5 x \right)} dx$ 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:
    
    $\text{False}$
    1. 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:
  
  $\frac{e^{\sin{\left(5 x \right)}}}{5}$
Method #2
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{e^{\sin{\left(u \right)}} \cos{\left(u \right)}}{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int e^{\sin{\left(u \right)}} \cos{\left(u \right)}\, du = \frac{\int e^{\sin{\left(u \right)}} \cos{\left(u \right)}\, du}{5}$
    1. Let $u = \sin{\left(u \right)}$ .
      
      Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
      
      $\int e^{u}\, du$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now substitute $u$ back in:
      
      $e^{\sin{\left(u \right)}}$
    So, the result is: $\frac{e^{\sin{\left(u \right)}}}{5}$
  Now substitute $u$ back in:
  
  $\frac{e^{\sin{\left(5 x \right)}}}{5}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                     
 |                              sin(5*x)
 |  sin(5*x)                   e        
 | E        *cos(5*x) dx = C + ---------
 |                                 5    
/

$$\int e^{\sin{\left(5 x \right)}} \cos{\left(5 x \right)}\, dx = C + \frac{e^{\sin{\left(5 x \right)}}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       sin(5)
  1   e      
- - + -------
  5      5

$$- \frac{1}{5} + \frac{1}{5 e^{- \sin{\left(5 \right)}}}$$

       sin(5)
  1   e      
- - + -------
  5      5

$$- \frac{1}{5} + \frac{1}{5 e^{- \sin{\left(5 \right)}}}$$

-1/5 + exp(sin(5))/5

Numerical answer [src]

-0.123339000965546

-0.123339000965546

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

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Identical expressions

Integral of (e^sin5x)*cos5x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2