Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x*e^(3*x)*dx
Integral of sin^-1(x)
Integral of 1/(2x-1)
Integral of 1/x(x+1)
Derivative of:
cos^2(4x)
Identical expressions
cos^ two (4x)
co sinus of e of squared (4x)
co sinus of e of to the power of two (4x)
cos2(4x)
cos24x
cos²(4x)
cos to the power of 2(4x)
cos^24x
cos^2(4x)dx
Similar expressions
cos^24x

Solving integrals/ cos^2(4x)

Integral of cos^2(4x) dx

The solution

You have entered [src]

  1             
  /             
 |              
 |     2        
 |  cos (4*x) dx
 |              
/               
0

$$\int\limits_{0}^{1} \cos^{2}{\left(4 x \right)}\, dx$$

Integral(cos(4*x)^2, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\cos^{2}{\left(4 x \right)} = \frac{\cos{\left(8 x \right)}}{2} + \frac{1}{2}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cos{\left(8 x \right)}}{2}\, dx = \frac{\int \cos{\left(8 x \right)}\, dx}{2}$
  1. Let $u = 8 x$ .
    
    Then let $du = 8 dx$ and substitute $\frac{du}{8}$ :
    
    $\int \frac{\cos{\left(u \right)}}{8}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{8}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{8}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(8 x \right)}}{8}$
  So, the result is: $\frac{\sin{\left(8 x \right)}}{16}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dx = \frac{x}{2}$
The result is: $\frac{x}{2} + \frac{\sin{\left(8 x \right)}}{16}$
Add the constant of integration:

$\frac{x}{2} + \frac{\sin{\left(8 x \right)}}{16}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} + \frac{\sin{\left(8 x \right)}}{16}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                                
 |    2               x   sin(8*x)
 | cos (4*x) dx = C + - + --------
 |                    2      16   
/

$$\int \cos^{2}{\left(4 x \right)}\, dx = C + \frac{x}{2} + \frac{\sin{\left(8 x \right)}}{16}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(4)*sin(4)
- + -------------
2         8

$$\frac{\sin{\left(4 \right)} \cos{\left(4 \right)}}{8} + \frac{1}{2}$$

1   cos(4)*sin(4)
- + -------------
2         8

$$\frac{\sin{\left(4 \right)} \cos{\left(4 \right)}}{8} + \frac{1}{2}$$

1/2 + cos(4)*sin(4)/8

Numerical answer [src]

0.561834890413961

0.561834890413961

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^2(4x) dx

Limits of integration:

The graph:

Piecewise:

The solution