Integral of sec dx

The solution

You have entered [src]

  1          
  /          
 |           
 |  sec(x) dx
 |           
/            
0

$$\int\limits_{0}^{1} \sec{\left(x \right)}\, dx$$

Integral(sec(x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sec{\left(x \right)} = \frac{\tan{\left(x \right)} \sec{\left(x \right)} + \sec^{2}{\left(x \right)}}{\tan{\left(x \right)} + \sec{\left(x \right)}}$
Let $u = \tan{\left(x \right)} + \sec{\left(x \right)}$ .

Then let $du = \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right) dx$ and substitute $du$ :

$\int \frac{1}{u}\, du$
1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
Now substitute $u$ back in:

$\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$
Add the constant of integration:

$\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                                     
 | sec(x) dx = C + log(sec(x) + tan(x))
 |                                     
/

$$\int \sec{\left(x \right)}\, dx = C + \log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(1 + sin(1))   log(1 - sin(1))
--------------- - ---------------
       2                 2

$$\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2}$$

log(1 + sin(1))   log(1 - sin(1))
--------------- - ---------------
       2                 2

$$\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2}$$

log(1 + sin(1))/2 - log(1 - sin(1))/2

Numerical answer [src]

1.22619117088352

1.22619117088352

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of sec dx

Limits of integration:

The graph:

Piecewise:

The solution