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Integral of d{x}:
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Integral of x⁴
Integral of x^2*ln(x)
Identical expressions
cos5x^ two
co sinus of e of 5x squared
co sinus of e of 5x to the power of two
cos5x2
cos5x²
cos5x to the power of 2
cos5x^2dx
Similar expressions
(3cos(5x))^2
sin(5x)^2×cos(5x)^2

Solving integrals/ cos5x^2

Integral of cos5x^2 dx

The solution

You have entered [src]

  1             
  /             
 |              
 |     2        
 |  cos (5*x) dx
 |              
/               
0

$$\int\limits_{0}^{1} \cos^{2}{\left(5 x \right)}\, dx$$

Simplify

Detail solution

Rewrite the integrand:

$\cos^{2}{\left(5 x \right)} = \frac{\cos{\left(10 x \right)}}{2} + \frac{1}{2}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cos{\left(10 x \right)}}{2}\, dx = \frac{\int \cos{\left(10 x \right)}\, dx}{2}$
  1. Let $u = 10 x$ .
    
    Then let $du = 10 dx$ and substitute $\frac{du}{10}$ :
    
    $\int \frac{\cos{\left(u \right)}}{100}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{10}\, du = \frac{\int \cos{\left(u \right)}\, du}{10}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{10}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(10 x \right)}}{10}$
  So, the result is: $\frac{\sin{\left(10 x \right)}}{20}$
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: $\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}$
Add the constant of integration:

$\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                                 
 |    2               x   sin(10*x)
 | cos (5*x) dx = C + - + ---------
 |                    2       20   
/

$${{{{\sin \left(10\,x\right)}\over{2}}+5\,x}\over{10}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(5)*sin(5)
- + -------------
2         10

$${{\sin 10+10}\over{20}}$$

1   cos(5)*sin(5)
- + -------------
2         10

$$\frac{\sin{\left(5 \right)} \cos{\left(5 \right)}}{10} + \frac{1}{2}$$

Simplify

Numerical answer [src]

0.472798944455532

0.472798944455532

The graph

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

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Identical expressions

Similar expressions

Integral of cos5x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution