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Identical expressions
sin(2x)cos(x)+ zero .5x
sinus of (2x) co sinus of e of (x) plus 0.5x
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sin2xcosx+0.5x
sin(2x)cos(x)+0.5xdx
Similar expressions
sin(2x)cos(x)-0.5x
sin(2x)cosx+0.5x

Solving integrals/ sin(2x)cos(x)+0.5x

Integral of sin(2x)cos(x)+0.5x dx

The solution

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  4                         
  /                         
 |                          
 |  /                  x\   
 |  |sin(2*x)*cos(x) + -| dx
 |  \                  2/   
 |                          
/                           
0

$$\int\limits_{0}^{4} \left(\frac{x}{2} + \sin{\left(2 x \right)} \cos{\left(x \right)}\right)\, dx$$

Integral(sin(2*x)*cos(x) + x/2, (x, 0, 4))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{2}\, dx = \frac{\int x\, dx}{2}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $\frac{x^{2}}{4}$
1. There are multiple ways to do this integral.
  Method #1
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{3}{\left(x \right)}}{3}$
    So, the result is: $- \frac{2 \cos^{3}{\left(x \right)}}{3}$
  Method #2
  1. Rewrite the integrand:
    
    $\sin{\left(2 x \right)} \cos{\left(x \right)} = 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u^{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{3}{\left(x \right)}}{3}$
    So, the result is: $- \frac{2 \cos^{3}{\left(x \right)}}{3}$
The result is: $\frac{x^{2}}{4} - \frac{2 \cos^{3}{\left(x \right)}}{3}$
Add the constant of integration:

$\frac{x^{2}}{4} - \frac{2 \cos^{3}{\left(x \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{x^{2}}{4} - \frac{2 \cos^{3}{\left(x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                     3       2
 | /                  x\          2*cos (x)   x 
 | |sin(2*x)*cos(x) + -| dx = C - --------- + --
 | \                  2/              3       4 
 |                                              
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

14   2*cos(4)*cos(8)   sin(4)*sin(8)
-- - --------------- - -------------
3           3                3

$$-{{\cos 12+3\,\cos 4-28}\over{6}}$$

14   2*cos(4)*cos(8)   sin(4)*sin(8)
-- - --------------- - -------------
3           3                3

$$- \frac{2 \cos{\left(4 \right)} \cos{\left(8 \right)}}{3} - \frac{\sin{\left(4 \right)} \sin{\left(8 \right)}}{3} + \frac{14}{3}$$

Упростить

Numerical answer [src]

4.85284615064306

4.85284615064306

The graph

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

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Identical expressions

Similar expressions

Integral of sin(2x)cos(x)+0.5x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2