Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of sqrt(4x^2-9)
Integral of (2x+7)^8
Integral of tan((pi)(x)/2)
Integral of sqrt(x)*sin(x)
Identical expressions
tan((pi)(x)/ two)
tangent of (( Pi )(x) divide by 2)
tangent of (( Pi )(x) divide by two)
tanpix/2
tan((pi)(x) divide by 2)
tan((pi)(x)/2)dx

Solving integrals/ tan((pi)(x)/2)

Integral of tan((pi)(x)/2) dx

The solution

You have entered [src]

  1             
  /             
 |              
 |     /pi*x\   
 |  tan|----| dx
 |     \ 2  /   
 |              
/               
0

$$\int\limits_{0}^{1} \tan{\left(\frac{\pi x}{2} \right)}\, dx$$

Integral(tan(pi*x/2), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\tan{\left(\frac{\pi x}{2} \right)} = \frac{\sin{\left(\frac{\pi x}{2} \right)}}{\cos{\left(\frac{\pi x}{2} \right)}}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \cos{\left(\frac{\pi x}{2} \right)}$ .
  
  Then let $du = - \frac{\pi \sin{\left(\frac{\pi x}{2} \right)} dx}{2}$ and substitute $- \frac{2 du}{\pi}$ :
  
  $\int \frac{4}{\pi^{2} u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{2}{u \pi}\right)\, du = - \frac{2 \int \frac{1}{u}\, du}{\pi}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $- \frac{2 \log{\left(u \right)}}{\pi}$
  Now substitute $u$ back in:
  
  $- \frac{2 \log{\left(\cos{\left(\frac{\pi x}{2} \right)} \right)}}{\pi}$
Method #2
1. Let $u = \frac{\pi x}{2}$ .
  
  Then let $du = \frac{\pi dx}{2}$ and substitute $\frac{2 du}{\pi}$ :
  
  $\int \frac{4 \sin{\left(u \right)}}{\pi^{2} \cos{\left(u \right)}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2 \sin{\left(u \right)}}{\pi \cos{\left(u \right)}}\, du = \frac{2 \int \frac{\sin{\left(u \right)}}{\cos{\left(u \right)}}\, du}{\pi}$
    1. Let $u = \cos{\left(u \right)}$ .
      
      Then let $du = - \sin{\left(u \right)} du$ and substitute $- du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \log{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- \log{\left(\cos{\left(u \right)} \right)}$
    So, the result is: $- \frac{2 \log{\left(\cos{\left(u \right)} \right)}}{\pi}$
  Now substitute $u$ back in:
  
  $- \frac{2 \log{\left(\cos{\left(\frac{\pi x}{2} \right)} \right)}}{\pi}$
Add the constant of integration:

$- \frac{2 \log{\left(\cos{\left(\frac{\pi x}{2} \right)} \right)}}{\pi}+ \mathrm{constant}$

The answer is:

$- \frac{2 \log{\left(\cos{\left(\frac{\pi x}{2} \right)} \right)}}{\pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                        /   /pi*x\\
 |                    2*log|cos|----||
 |    /pi*x\               \   \ 2  //
 | tan|----| dx = C - ----------------
 |    \ 2  /                 pi       
 |                                    
/

$${{2\,\log \sec \left({{\pi\,x}\over{2}}\right)}\over{\pi}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$-{{2\,\log \left| \cos \left({{\pi}\over{2}}\right)\right| }\over{ \pi}}$$

oo

$$\infty$$

Упростить

Numerical answer [src]

27.7822258859063

27.7822258859063

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of tan((pi)(x)/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2