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Integral of 1/tgx*cos^2x dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

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  1           
  /           
 |            
 |     2      
 |  cos (x)   
 |  ------- dx
 |   tan(x)   
 |            
/             
0             
$$\int\limits_{0}^{1} \frac{\cos^{2}{\left(x \right)}}{\tan{\left(x \right)}}\, dx$$
Integral(cos(x)^2/tan(x), (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

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

          1. The integral of is .

          So, the result is:

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

          1. The integral of is when :

          So, the result is:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Let .

      Then let and substitute :

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

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

          1. The integral of is .

          So, the result is:

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

          1. The integral of is when :

          So, the result is:

        The result is:

      Now substitute back in:

    Method #3

    1. Rewrite the integrand:

    2. Let .

      Then let and substitute :

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

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

          1. The integral of is .

          So, the result is:

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

          1. The integral of is when :

          So, the result is:

        The result is:

      Now substitute back in:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                             
 |                                                              
 |    2                            /        2   \      /   2   \
 | cos (x)              1       log\-1 + sec (x)/   log\sec (x)/
 | ------- dx = C + --------- + ----------------- - ------------
 |  tan(x)               2              2                2      
 |                  2*sec (x)                                   
/                                                               
$$\int \frac{\cos^{2}{\left(x \right)}}{\tan{\left(x \right)}}\, dx = C + \frac{\log{\left(\sec^{2}{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{2} + \frac{1}{2 \sec^{2}{\left(x \right)}}$$
The graph
The answer [src]
oo
$$\infty$$
=
=
oo
$$\infty$$
oo
Numerical answer [src]
43.563805678587
43.563805678587

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