Mister Exam

Integral of cosecx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1          
  /          
 |           
 |  csc(x) dx
 |           
/            
0            
01csc(x)dx\int\limits_{0}^{1} \csc{\left(x \right)}\, dx
Integral(csc(x), (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

    csc(x)=cot(x)csc(x)+csc2(x)cot(x)+csc(x)\csc{\left(x \right)} = \frac{\cot{\left(x \right)} \csc{\left(x \right)} + \csc^{2}{\left(x \right)}}{\cot{\left(x \right)} + \csc{\left(x \right)}}

  2. Let u=cot(x)+csc(x)u = \cot{\left(x \right)} + \csc{\left(x \right)}.

    Then let du=(cot2(x)cot(x)csc(x)1)dxdu = \left(- \cot^{2}{\left(x \right)} - \cot{\left(x \right)} \csc{\left(x \right)} - 1\right) dx and substitute du- du:

    (1u)du\int \left(- \frac{1}{u}\right)\, du

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

      1udu=1udu\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du

      1. The integral of 1u\frac{1}{u} is log(u)\log{\left(u \right)}.

      So, the result is: log(u)- \log{\left(u \right)}

    Now substitute uu back in:

    log(cot(x)+csc(x))- \log{\left(\cot{\left(x \right)} + \csc{\left(x \right)} \right)}

  3. Add the constant of integration:

    log(cot(x)+csc(x))+constant- \log{\left(\cot{\left(x \right)} + \csc{\left(x \right)} \right)}+ \mathrm{constant}


The answer is:

log(cot(x)+csc(x))+constant- \log{\left(\cot{\left(x \right)} + \csc{\left(x \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                    
 |                                     
 | csc(x) dx = C - log(cot(x) + csc(x))
 |                                     
/                                      
csc(x)dx=Clog(cot(x)+csc(x))\int \csc{\left(x \right)}\, dx = C - \log{\left(\cot{\left(x \right)} + \csc{\left(x \right)} \right)}
The graph
0.001.000.100.200.300.400.500.600.700.800.90010000
The answer [src]
     pi*I
oo + ----
      2  
+iπ2\infty + \frac{i \pi}{2}
=
=
     pi*I
oo + ----
      2  
+iπ2\infty + \frac{i \pi}{2}
oo + pi*i/2
Numerical answer [src]
44.1790108686112
44.1790108686112
The graph
Integral of cosecx dx

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