Mister Exam

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Integral of d{x}:
Integral of -e^x
Integral of sqrt(x^2+1)/x^3
Integral of 2x*e^x^2
Integral of dx/(9*x^2-1)
Identical expressions
sec^2y/(tgy)
sec squared y divide by (tgy)
sec2y/(tgy)
sec2y/tgy
sec²y/(tgy)
sec to the power of 2y/(tgy)
sec^2y/tgy
sec^2y divide by (tgy)

Integral of sec^2y/(tgy) dx

The solution

You have entered [src]

  1           
  /           
 |            
 |     2      
 |  sec (y)   
 |  ------- dy
 |   tan(y)   
 |            
/             
0

$$\int\limits_{0}^{1} \frac{\sec^{2}{\left(y \right)}}{\tan{\left(y \right)}}\, dy$$

Integral(sec(y)^2/tan(y), (y, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \tan{\left(y \right)}$ .
  
  Then let $du = \left(\tan^{2}{\left(y \right)} + 1\right) dy$ 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:
  
  $\log{\left(\tan{\left(y \right)} \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\sec^{2}{\left(y \right)}}{\tan{\left(y \right)}} = \frac{\tan{\left(y \right)} \sec^{2}{\left(y \right)}}{\sec^{2}{\left(y \right)} - 1}$
2. Let $u = \sec^{2}{\left(y \right)} - 1$ .
  
  Then let $du = 2 \tan{\left(y \right)} \sec^{2}{\left(y \right)} dy$ 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:
  
  $\frac{\log{\left(\sec^{2}{\left(y \right)} - 1 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                            
 |                             
 |    2                        
 | sec (y)                     
 | ------- dy = C + log(tan(y))
 |  tan(y)                     
 |                             
/

$$\int \frac{\sec^{2}{\left(y \right)}}{\tan{\left(y \right)}}\, dy = C + \log{\left(\tan{\left(y \right)} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     pi*I
oo - ----
      2

$$\infty - \frac{i \pi}{2}$$

     pi*I
oo - ----
      2

$$\infty - \frac{i \pi}{2}$$

oo - pi*i/2

Numerical answer [src]

44.5334688581098

44.5334688581098

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

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Identical expressions

Integral of sec^2y/(tgy) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2