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Solving integrals/ tanxsec^2x

Integral of tanxsec^2x dx

The solution

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 pi                  
 --                  
 4                   
  /                  
 |                   
 |            2      
 |  tan(x)*sec (x) dx
 |                   
/                    
0

$$\int\limits_{0}^{\frac{\pi}{4}} \tan{\left(x \right)} \sec^{2}{\left(x \right)}\, dx$$

Integral(tan(x)*sec(x)^2, (x, 0, pi/4))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                               
 |                            2   
 |           2             tan (x)
 | tan(x)*sec (x) dx = C + -------
 |                            2   
/

$$\int \tan{\left(x \right)} \sec^{2}{\left(x \right)}\, dx = C + \frac{\tan^{2}{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/2

$$\frac{1}{2}$$

1/2

$$\frac{1}{2}$$

1/2

Numerical answer [src]

0.5

0.5

The graph

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

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How to use it?

Identical expressions

Integral of tanxsec^2x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2