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(sec(x))^3

Integral of (sec(x))^3 dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |     3      
 |  sec (x) dx
 |            
/             
0             
$$\int\limits_{0}^{1} \sec^{3}{\left(x \right)}\, dx$$
Integral(sec(x)^3, (x, 0, 1))
The answer (Indefinite) [src]
  /                                                                    
 |                                                                     
 |    3             log(-1 + sin(x))   log(1 + sin(x))       sin(x)    
 | sec (x) dx = C - ---------------- + --------------- - --------------
 |                         4                  4                    2   
/                                                        -2 + 2*sin (x)
$$\int \sec^{3}{\left(x \right)}\, dx = C - \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{4} - \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - 2}$$
The graph
The answer [src]
  log(1 - sin(1))   log(1 + sin(1))       sin(1)    
- --------------- + --------------- - --------------
         4                 4                    2   
                                      -2 + 2*sin (1)
$$\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{4} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{4} - \frac{\sin{\left(1 \right)}}{-2 + 2 \sin^{2}{\left(1 \right)}}$$
=
=
  log(1 - sin(1))   log(1 + sin(1))       sin(1)    
- --------------- + --------------- - --------------
         4                 4                    2   
                                      -2 + 2*sin (1)
$$\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{4} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{4} - \frac{\sin{\left(1 \right)}}{-2 + 2 \sin^{2}{\left(1 \right)}}$$
-log(1 - sin(1))/4 + log(1 + sin(1))/4 - sin(1)/(-2 + 2*sin(1)^2)
Numerical answer [src]
2.05433293325625
2.05433293325625
The graph
Integral of (sec(x))^3 dx

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