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Similar expressions
x^4-3sin(3x)

Integral of x^4+3sin(3x) dx

The solution

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

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

Integral(x^4 + 3*sin(3*x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{4}\, dx = \frac{x^{5}}{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3 \sin{\left(3 x \right)}\, dx = 3 \int \sin{\left(3 x \right)}\, dx$
  1. Let $u = 3 x$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{\sin{\left(u \right)}}{3}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
      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:
    
    $- \frac{\cos{\left(3 x \right)}}{3}$
  So, the result is: $- \cos{\left(3 x \right)}$
The result is: $\frac{x^{5}}{5} - \cos{\left(3 x \right)}$
Add the constant of integration:

$\frac{x^{5}}{5} - \cos{\left(3 x \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{5}}{5} - \cos{\left(3 x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int \left(x^{4} + 3 \sin{\left(3 x \right)}\right)\, dx = C + \frac{x^{5}}{5} - \cos{\left(3 x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

6/5 - cos(3)

$$\frac{6}{5} - \cos{\left(3 \right)}$$

6/5 - cos(3)

$$\frac{6}{5} - \cos{\left(3 \right)}$$

6/5 - cos(3)

Numerical answer [src]

2.18999249660045

2.18999249660045

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

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Integral of x^4+3sin(3x) dx

Limits of integration:

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Piecewise:

The solution