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Solving integrals/ (cos(x^2)+sin(x^2))*x

Integral of (cos(x^2)+sin(x^2))*x dx

The solution

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  1                         
  /                         
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 |  /   / 2\      / 2\\     
 |  \cos\x / + sin\x //*x dx
 |                          
/                           
0

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

Integral((cos(x^2) + sin(x^2))*x, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = x^{2}$ .
  
  Then let $du = 2 x dx$ and substitute $du$ :
  
  $\int \left(\frac{\sin{\left(u \right)}}{2} + \frac{\cos{\left(u \right)}}{2}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(u \right)}}{2}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
    The result is: $\frac{\sin{\left(u \right)}}{2} - \frac{\cos{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(x^{2} \right)}}{2} - \frac{\cos{\left(x^{2} \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $x \left(\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}\right) = x \sin{\left(x^{2} \right)} + x \cos{\left(x^{2} \right)}$
2. Integrate term-by-term:
  1. Let $u = x^{2}$ .
    
    Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\sin{\left(u \right)}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $- \frac{\cos{\left(x^{2} \right)}}{2}$
  1. Let $u = x^{2}$ .
    
    Then let $du = 2 x dx$ 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:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      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:
    
    $\frac{\sin{\left(x^{2} \right)}}{2}$
  The result is: $\frac{\sin{\left(x^{2} \right)}}{2} - \frac{\cos{\left(x^{2} \right)}}{2}$
Method #3
1. Rewrite the integrand:
  
  $x \left(\sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}\right) = x \sin{\left(x^{2} \right)} + x \cos{\left(x^{2} \right)}$
2. Integrate term-by-term:
  1. Let $u = x^{2}$ .
    
    Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\sin{\left(u \right)}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $- \frac{\cos{\left(x^{2} \right)}}{2}$
  1. Let $u = x^{2}$ .
    
    Then let $du = 2 x dx$ 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:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      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:
    
    $\frac{\sin{\left(x^{2} \right)}}{2}$
  The result is: $\frac{\sin{\left(x^{2} \right)}}{2} - \frac{\cos{\left(x^{2} \right)}}{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

0.650584339469878

0.650584339469878

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

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Identical expressions

Similar expressions

Integral of (cos(x^2)+sin(x^2))*x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3