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Identical expressions
(x^ three)/ three +(sin five x)/5
(x cubed ) divide by 3 plus ( sinus of 5x) divide by 5
(x to the power of three) divide by three plus ( sinus of five x) divide by 5
(x3)/3+(sin5x)/5
x3/3+sin5x/5
(x³)/3+(sin5x)/5
(x to the power of 3)/3+(sin5x)/5
x^3/3+sin5x/5
(x^3) divide by 3+(sin5x) divide by 5
(x^3)/3+(sin5x)/5dx
Similar expressions
(x^3)/3-(sin5x)/5

Solving integrals/ (x^3)/3+(sin5x)/5

Integral of (x^3)/3+(sin5x)/5 dx

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |  / 3           \   
 |  |x    sin(5*x)|   
 |  |-- + --------| dx
 |  \3       5    /   
 |                    
/                     
0

\int\limits_{0}^{1} \left(\frac{x^{3}}{3} + \frac{\sin{\left(5 x \right)}}{5}\right)\, dx

Integral(x^3/3 + sin(5*x)/5, (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:
  
  $\int \frac{x^{3}}{3}\, dx = \frac{\int x^{3}\, dx}{3}$
  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: $\frac{x^{4}}{12}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sin{\left(5 x \right)}}{5}\, dx = \frac{\int \sin{\left(5 x \right)}\, dx}{5}$
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{\sin{\left(u \right)}}{25}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      1. The integral of sine is negative cosine:
        
        $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $- \frac{\cos{\left(5 x \right)}}{5}$
  So, the result is: $- \frac{\cos{\left(5 x \right)}}{25}$
The result is: $\frac{x^{4}}{12} - \frac{\cos{\left(5 x \right)}}{25}$
Add the constant of integration:

$\frac{x^{4}}{12} - \frac{\cos{\left(5 x \right)}}{25}+ \mathrm{constant}$

The answer is:

\frac{x^{4}}{12} - \frac{\cos{\left(5 x \right)}}{25}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                      
 |                                       
 | / 3           \                      4
 | |x    sin(5*x)|          cos(5*x)   x 
 | |-- + --------| dx = C - -------- + --
 | \3       5    /             25      12
 |                                       
/

{{x^4}\over{12}}-{{\cos \left(5\,x\right)}\over{25}}

Simplify

The graph

Plot the graph f(x)

The answer [src]

 37   cos(5)
--- - ------
300     25

-{{12\,\cos 5-37}\over{300}}

 37   cos(5)
--- - ------
300     25

- \frac{\cos{\left(5 \right)}}{25} + \frac{37}{300}

Simplify

Numerical answer [src]

0.111986845914804

0.111986845914804

The graph

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

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Integral of (x^3)/3+(sin5x)/5 dx

Limits of integration:

The graph:

Piecewise:

The solution