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Identical expressions
(x+tgx- one /sin2x)*dx
(x plus tgx minus 1 divide by sinus of 2x) multiply by dx
(x plus tgx minus one divide by sinus of 2x) multiply by dx
(x+tgx-1/sin2x)dx
x+tgx-1/sin2xdx
(x+tgx-1 divide by sin2x)*dx
Similar expressions
(x-tgx-1/sin2x)*dx
(x+tgx+1/sin2x)*dx

Integral of (x+tgx-1/sin2x)*dx dx

The solution

You have entered [src]

  1                               
  /                               
 |                                
 |  /                  1    \     
 |  |x + tan(x) - 1*--------|*1 dx
 |  \               sin(2*x)/     
 |                                
/                                 
0

$$\int\limits_{0}^{1} \left(x + \tan{\left(x \right)} - 1 \cdot \frac{1}{\sin{\left(2 x \right)}}\right) 1\, dx$$

Integral((x + tan(x) - 1/sin(2*x))*1, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(x + \tan{\left(x \right)} - 1 \cdot \frac{1}{\sin{\left(2 x \right)}}\right) 1 = \frac{x \sin{\left(2 x \right)} + \sin{\left(2 x \right)} \tan{\left(x \right)} - 1}{\sin{\left(2 x \right)}}$
2. Rewrite the integrand:
  
  $\frac{x \sin{\left(2 x \right)} + \sin{\left(2 x \right)} \tan{\left(x \right)} - 1}{\sin{\left(2 x \right)}} = x + \tan{\left(x \right)} - \frac{1}{2 \sin{\left(x \right)} \cos{\left(x \right)}}$
3. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  1. Rewrite the integrand:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- \log{\left(\cos{\left(x \right)} \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{2 \sin{\left(x \right)} \cos{\left(x \right)}}\right)\, dx = - \frac{\int \frac{1}{\sin{\left(x \right)} \cos{\left(x \right)}}\, dx}{2}$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $- \frac{\log{\left(\sin^{2}{\left(x \right)} - 1 \right)}}{2} + \log{\left(\sin{\left(x \right)} \right)}$
    So, the result is: $\frac{\log{\left(\sin^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{\log{\left(\sin{\left(x \right)} \right)}}{2}$
  The result is: $\frac{x^{2}}{2} + \frac{\log{\left(\sin^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{\log{\left(\sin{\left(x \right)} \right)}}{2} - \log{\left(\cos{\left(x \right)} \right)}$
Method #2
1. Rewrite the integrand:
  
  $\left(x + \tan{\left(x \right)} - 1 \cdot \frac{1}{\sin{\left(2 x \right)}}\right) 1 = x + \tan{\left(x \right)} - \frac{1}{2 \sin{\left(x \right)} \cos{\left(x \right)}}$
2. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  1. Rewrite the integrand:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- \log{\left(\cos{\left(x \right)} \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{2 \sin{\left(x \right)} \cos{\left(x \right)}}\right)\, dx = - \frac{\int \frac{1}{\sin{\left(x \right)} \cos{\left(x \right)}}\, dx}{2}$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $- \frac{\log{\left(\sin^{2}{\left(x \right)} - 1 \right)}}{2} + \log{\left(\sin{\left(x \right)} \right)}$
    So, the result is: $\frac{\log{\left(\sin^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{\log{\left(\sin{\left(x \right)} \right)}}{2}$
  The result is: $\frac{x^{2}}{2} + \frac{\log{\left(\sin^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{\log{\left(\sin{\left(x \right)} \right)}}{2} - \log{\left(\cos{\left(x \right)} \right)}$
Now simplify:

$\frac{x^{2}}{2} - \frac{\log{\left(\sin{\left(x \right)} \right)}}{2} - \log{\left(\cos{\left(x \right)} \right)} + \frac{\log{\left(- \cos^{2}{\left(x \right)} \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                                                       
 |                                       2                                  /        2   \
 | /                  1    \            x                  log(sin(x))   log\-1 + sin (x)/
 | |x + tan(x) - 1*--------|*1 dx = C + -- - log(cos(x)) - ----------- + -----------------
 | \               sin(2*x)/            2                       2                4        
 |                                                                                        
/

$$-{{{{\log \left(\cos \left(2\,x\right)-1\right)}\over{2}}-{{\log \left(\cos \left(2\,x\right)+1\right)}\over{2}}}\over{2}}+\log \sec x+{{x^2}\over{2}}$$

Simplify

The answer [src]

      pi*I
-oo - ----
       4

$${\it \%a}$$

      pi*I
-oo - ----
       4

$$-\infty - \frac{i \pi}{4}$$

Упростить

Numerical answer [src]

-21.1511079586689

-21.1511079586689

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

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Identical expressions

Similar expressions

Integral of (x+tgx-1/sin2x)*dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2