Integral of cos(x)*sin(x) dx

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

Integral(cos(x)*sin(x), (x, 0, 1))

Detail solution

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:
    
    $\int \left(- u\right)\, du = - \int u\, du$
    1. 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:
  
  $\frac{\sin^{2}{\left(x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

   2   
sin (1)
-------
   2

\frac{\sin^{2}{\left(1 \right)}}{2}

   2   
sin (1)
-------
   2

\frac{\sin^{2}{\left(1 \right)}}{2}

Упростить

Numerical answer [src]

0.354036709136786

0.354036709136786

The graph

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

Method #1