Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of 3/x
Integral of sin(log(x))/x^2
Integral of xarctan(2x)
Integral of x^2+y^2
Identical expressions
cos(two x-(pi/2))
co sinus of e of (2x minus ( Pi divide by 2))
co sinus of e of (two x minus ( Pi divide by 2))
cos2x-pi/2
cos(2x-(pi divide by 2))
cos(2x-(pi/2))dx
Similar expressions
cos(2x+(pi/2))

Solving integrals/ cos(2x-(pi/2))

Integral of cos(2x-(pi/2)) dx

The solution

You have entered [src]

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

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

Integral(cos(2*x - pi/2), (x, 0, 1))

Detail solution

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

$- \frac{\cos{\left(2 x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                          /      pi\
 |                        sin|2*x - --|
 |    /      pi\             \      2 /
 | cos|2*x - --| dx = C + -------------
 |    \      2 /                2      
 |                                     
/

$${{\sin \left(2\,x-{{\pi}\over{2}}\right)}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(2)
- - ------
2     2

$${{\sin \left({{\pi}\over{2}}\right)}\over{2}}-{{\sin \left({{\pi-4 }\over{2}}\right)}\over{2}}$$

1   cos(2)
- - ------
2     2

$$\frac{1}{2} - \frac{\cos{\left(2 \right)}}{2}$$

Упростить

Numerical answer [src]

0.708073418273571

0.708073418273571

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cos(2x-(pi/2)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2