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Integral of d{x}:
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Integral of 1/(x²+1)²
Identical expressions
y=4cosx+sin3x−8x.
y equally 4 co sinus of e of x plus sinus of 3x−8x.
y=4cosx+sin3x−8x.dx
Similar expressions
y=4cosx-sin3x−8x.

Solving integrals/ y=4cosx+sin3x−8x.

Integral of y=4cosx+sin3x−8x. dx

The solution

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  1                               
  /                               
 |                                
 |  (4*cos(x) + sin(3*x) - 8*x) dx
 |                                
/                                 
0

$$\int\limits_{0}^{1} \left(- 8 x + \sin{\left(3 x \right)} + 4 \cos{\left(x \right)}\right)\, dx$$

Integral(4*cos(x) + sin(3*x) - 8*x, (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:
  
  $\int \left(- 8 x\right)\, dx = - \int 8 x\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 8 x\, dx = 8 \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $4 x^{2}$
  So, the result is: $- 4 x^{2}$
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\sin{\left(u \right)}}{9}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{3}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(3 x \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 \cos{\left(x \right)}\, dx = 4 \int \cos{\left(x \right)}\, dx$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $4 \sin{\left(x \right)}$
The result is: $- 4 x^{2} + 4 \sin{\left(x \right)} - \frac{\cos{\left(3 x \right)}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                               
 |                                         2              cos(3*x)
 | (4*cos(x) + sin(3*x) - 8*x) dx = C - 4*x  + 4*sin(x) - --------
 |                                                           3    
/

$$-{{\cos \left(3\,x\right)}\over{3}}+4\,\sin x-4\,x^2$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  11              cos(3)
- -- + 4*sin(1) - ------
  3                 3

$$-{{\cos 3-12\,\sin 1+11}\over{3}}$$

  11              cos(3)
- -- + 4*sin(1) - ------
  3                 3

$$- \frac{11}{3} - \frac{\cos{\left(3 \right)}}{3} + 4 \sin{\left(1 \right)}$$

Упростить

Numerical answer [src]

0.0292147714317345

0.0292147714317345

The graph

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

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Identical expressions

Similar expressions

Integral of y=4cosx+sin3x−8x. dx

Limits of integration:

The graph:

Piecewise:

The solution