Integral of sec(2x) dx

The solution

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 |  sec(2*x) dx
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$$\int\limits_{0}^{1} \sec{\left(2 x \right)}\, dx$$

Integral(sec(2*x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sec{\left(2 x \right)} = \frac{\tan{\left(2 x \right)} \sec{\left(2 x \right)} + \sec^{2}{\left(2 x \right)}}{\tan{\left(2 x \right)} + \sec{\left(2 x \right)}}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \tan{\left(2 x \right)} + \sec{\left(2 x \right)}$ .
  
  Then let $du = \left(2 \tan^{2}{\left(2 x \right)} + 2 \tan{\left(2 x \right)} \sec{\left(2 x \right)} + 2\right) dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{1}{2 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $\frac{\log{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)}}{2}$
Method #2
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{\sec{\left(u \right)}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sec{\left(u \right)}\, du = \frac{\int \sec{\left(u \right)}\, du}{2}$
    1. Rewrite the integrand:
      
      $\sec{\left(u \right)} = \frac{\tan{\left(u \right)} \sec{\left(u \right)} + \sec^{2}{\left(u \right)}}{\tan{\left(u \right)} + \sec{\left(u \right)}}$
    2. Let $u = \tan{\left(u \right)} + \sec{\left(u \right)}$ .
      
      Then let $du = \left(\tan^{2}{\left(u \right)} + \tan{\left(u \right)} \sec{\left(u \right)} + 1\right) du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(\tan{\left(u \right)} + \sec{\left(u \right)} \right)}$
    So, the result is: $\frac{\log{\left(\tan{\left(u \right)} + \sec{\left(u \right)} \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)}}{2}$
Add the constant of integration:

$\frac{\log{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          
 |                   log(sec(2*x) + tan(2*x))
 | sec(2*x) dx = C + ------------------------
 |                              2            
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$$\int \sec{\left(2 x \right)}\, dx = C + \frac{\log{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

-0.637807798738107

-0.637807798738107

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

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Identical expressions

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Integral of sec(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2