Mister Exam

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Similar expressions
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Integral of (cos3x-2x^3) dx

The solution

You have entered [src]

  1                     
  /                     
 |                      
 |  /              3\   
 |  \cos(3*x) - 2*x / dx
 |                      
/                       
0

$$\int\limits_{0}^{1} \left(- 2 x^{3} + \cos{\left(3 x \right)}\right)\, dx$$

Integral(cos(3*x) - 2*x^3, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 2 x^{3}\right)\, dx = - 2 \int x^{3}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{3}\, dx = \frac{x^{4}}{4}$
  So, the result is: $- \frac{x^{4}}{2}$
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\cos{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(3 x \right)}}{3}$
The result is: $- \frac{x^{4}}{2} + \frac{\sin{\left(3 x \right)}}{3}$
Add the constant of integration:

$- \frac{x^{4}}{2} + \frac{\sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{x^{4}}{2} + \frac{\sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                        
 |                             4           
 | /              3\          x    sin(3*x)
 | \cos(3*x) - 2*x / dx = C - -- + --------
 |                            2       3    
/

$$\int \left(- 2 x^{3} + \cos{\left(3 x \right)}\right)\, dx = C - \frac{x^{4}}{2} + \frac{\sin{\left(3 x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   sin(3)
- - + ------
  2     3

$$- \frac{1}{2} + \frac{\sin{\left(3 \right)}}{3}$$

  1   sin(3)
- - + ------
  2     3

$$- \frac{1}{2} + \frac{\sin{\left(3 \right)}}{3}$$

-1/2 + sin(3)/3

Numerical answer [src]

-0.452959997313378

-0.452959997313378

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

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Integral of (cos3x-2x^3) dx

Limits of integration:

The graph:

Piecewise:

The solution