Integral of ecosx*sinxdx dx

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 POST_GRBEK_SMALL_pi                    
 -------------------                    
          2                             
          /                             
         |                              
         |          e*cos(x)*sin(x)*1 dx
         |                              
        /                               
        0

$$\int\limits_{0}^{\frac{POST_{GRBEK SMALL \pi}}{2}} e \cos{\left(x \right)} \sin{\left(x \right)} 1\, dx$$

Integral(E*cos(x)*sin(x)*1, (x, 0, POST_GRBEK_SMALL_pi/2))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                2   
 |                            e*cos (x)
 | e*cos(x)*sin(x)*1 dx = C - ---------
 |                                2    
/

$$\int e \cos{\left(x \right)} \sin{\left(x \right)} 1\, dx = C - \frac{e \cos^{2}{\left(x \right)}}{2}$$

Simplify

The answer [src]

     2/POST_GRBEK_SMALL_pi\
e*sin |-------------------|
      \         2         /
---------------------------
             2

$$\frac{e \sin^{2}{\left(\frac{POST_{GRBEK SMALL \pi}}{2} \right)}}{2}$$

=

     2/POST_GRBEK_SMALL_pi\
e*sin |-------------------|
      \         2         /
---------------------------
             2

$$\frac{e \sin^{2}{\left(\frac{POST_{GRBEK SMALL \pi}}{2} \right)}}{2}$$

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Integral of ecosx*sinxdx dx

Limits of integration:

The graph:

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The solution

Method #1

Method #2