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sinx/cosx*cosx

Integral of sinx/cosx*cosx dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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  1                 
  /                 
 |                  
 |  sin(x)*cos(x)   
 |  ------------- dx
 |      cos(x)      
 |                  
/                   
0                   
01sin(x)cos(x)cos(x)dx\int\limits_{0}^{1} \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)}}\, dx
Integral(sin(x)*cos(x)/cos(x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=1cos(x)u = \frac{1}{\cos{\left(x \right)}}.

      Then let du=sin(x)dxcos2(x)du = \frac{\sin{\left(x \right)} dx}{\cos^{2}{\left(x \right)}} and substitute dudu:

      1u2du\int \frac{1}{u^{2}}\, du

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

        1u2du=1u\int \frac{1}{u^{2}}\, du = - \frac{1}{u}

      Now substitute uu back in:

      cos(x)- \cos{\left(x \right)}

    Method #2

    1. Let u=cos(x)u = \cos{\left(x \right)}.

      Then let du=sin(x)dxdu = - \sin{\left(x \right)} dx and substitute du- du:

      1du\int 1\, du

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

        (1)du=1du\int \left(-1\right)\, du = - \int 1\, du

        1. The integral of a constant is the constant times the variable of integration:

          1du=u\int 1\, du = u

        So, the result is: u- u

      Now substitute uu back in:

      cos(x)- \cos{\left(x \right)}

  2. Add the constant of integration:

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


The answer is:

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

The answer (Indefinite) [src]
  /                             
 |                              
 | sin(x)*cos(x)                
 | ------------- dx = C - cos(x)
 |     cos(x)                   
 |                              
/                               
sin(x)cos(x)cos(x)dx=Ccos(x)\int \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)}}\, dx = C - \cos{\left(x \right)}
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
The answer [src]
1 - cos(1)
1cos(1)1 - \cos{\left(1 \right)}
=
=
1 - cos(1)
1cos(1)1 - \cos{\left(1 \right)}
Numerical answer [src]
0.45969769413186
0.45969769413186
The graph
Integral of sinx/cosx*cosx dx

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