Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of sin
Integral of cos(x)sin⁴(x)
Integral of 3x^2-4x-2
Integral of cos^2tdt
Identical expressions
cos^2tdt
co sinus of e of squared tdt
cos2tdt
cos²tdt
cos to the power of 2tdt

Integral of cos^2tdt dx

The solution

You have entered [src]

  0           
  /           
 |            
 |     2      
 |  cos (t) dt
 |            
/             
pi            
--            
4

$$\int\limits_{\frac{\pi}{4}}^{0} \cos^{2}{\left(t \right)}\, dt$$

Integral(cos(t)^2, (t, pi/4, 0))

Detail solution

Rewrite the integrand:

$\cos^{2}{\left(t \right)} = \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cos{\left(2 t \right)}}{2}\, dt = \frac{\int \cos{\left(2 t \right)}\, dt}{2}$
  1. Let $u = 2 t$ .
    
    Then let $du = 2 dt$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\cos{\left(u \right)}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      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:
    
    $\frac{\sin{\left(2 t \right)}}{2}$
  So, the result is: $\frac{\sin{\left(2 t \right)}}{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dt = \frac{t}{2}$
The result is: $\frac{t}{2} + \frac{\sin{\left(2 t \right)}}{4}$
Add the constant of integration:

$\frac{t}{2} + \frac{\sin{\left(2 t \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{t}{2} + \frac{\sin{\left(2 t \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             
 |                              
 |    2             t   sin(2*t)
 | cos (t) dt = C + - + --------
 |                  2      4    
/

$$\int \cos^{2}{\left(t \right)}\, dt = C + \frac{t}{2} + \frac{\sin{\left(2 t \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   pi
- - - --
  4   8

$$- \frac{\pi}{8} - \frac{1}{4}$$

  1   pi
- - - --
  4   8

$$- \frac{\pi}{8} - \frac{1}{4}$$

-1/4 - pi/8

Numerical answer [src]

-0.642699081698724

-0.642699081698724

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

Other calculators

How to use it?

Identical expressions

Integral of cos^2tdt dx

Limits of integration:

The graph:

Piecewise:

The solution