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Identical expressions
(4x- two)*cos2(x)
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(4x minus two) multiply by co sinus of e of 2(x)
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Similar expressions
(4x-2)cos(2x)
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Integral of (4x-2)*cos2(x) dx

The solution

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0

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(4 x - 2\right) \cos^{2}{\left(x \right)} = 4 x \cos^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 x \cos^{2}{\left(x \right)}\, dx = 4 \int x \cos^{2}{\left(x \right)}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x^{2} \sin^{2}{\left(x \right)}}{4} + \frac{x^{2} \cos^{2}{\left(x \right)}}{4} + \frac{x \sin{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\cos^{2}{\left(x \right)}}{4}$
    So, the result is: $x^{2} \sin^{2}{\left(x \right)} + x^{2} \cos^{2}{\left(x \right)} + 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \cos^{2}{\left(x \right)}\right)\, dx = - 2 \int \cos^{2}{\left(x \right)}\, dx$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      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:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
      
      So, the result is: $\frac{\sin{\left(2 x \right)}}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
    So, the result is: $- x - \frac{\sin{\left(2 x \right)}}{2}$
  The result is: $x^{2} \sin^{2}{\left(x \right)} + x^{2} \cos^{2}{\left(x \right)} + 2 x \sin{\left(x \right)} \cos{\left(x \right)} - x - \frac{\sin{\left(2 x \right)}}{2} + \cos^{2}{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\left(4 x - 2\right) \cos^{2}{\left(x \right)} = 4 x \cos^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 x \cos^{2}{\left(x \right)}\, dx = 4 \int x \cos^{2}{\left(x \right)}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x^{2} \sin^{2}{\left(x \right)}}{4} + \frac{x^{2} \cos^{2}{\left(x \right)}}{4} + \frac{x \sin{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\cos^{2}{\left(x \right)}}{4}$
    So, the result is: $x^{2} \sin^{2}{\left(x \right)} + x^{2} \cos^{2}{\left(x \right)} + 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \cos^{2}{\left(x \right)}\right)\, dx = - 2 \int \cos^{2}{\left(x \right)}\, dx$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      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:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
      
      So, the result is: $\frac{\sin{\left(2 x \right)}}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
    So, the result is: $- x - \frac{\sin{\left(2 x \right)}}{2}$
  The result is: $x^{2} \sin^{2}{\left(x \right)} + x^{2} \cos^{2}{\left(x \right)} + 2 x \sin{\left(x \right)} \cos{\left(x \right)} - x - \frac{\sin{\left(2 x \right)}}{2} + \cos^{2}{\left(x \right)}$
Now simplify:

$x^{2} + x \sin{\left(2 x \right)} - x + \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{2} + \frac{1}{2}$
Add the constant of integration:

$x^{2} + x \sin{\left(2 x \right)} - x + \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{2} + \frac{1}{2}+ \mathrm{constant}$

The answer is:

$x^{2} + x \sin{\left(2 x \right)} - x + \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{2} + \frac{1}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$$-1 + \cos^{2}{\left(1 \right)} + \sin{\left(1 \right)} \cos{\left(1 \right)}$$

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

$$-1 + \cos^{2}{\left(1 \right)} + \sin{\left(1 \right)} \cos{\left(1 \right)}$$

-1 + cos(1)^2 + cos(1)*sin(1)

Numerical answer [src]

-0.25342470486073

-0.25342470486073

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

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How to use it?

Identical expressions

Similar expressions

Integral of (4x-2)*cos2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2