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Similar expressions
8x^3sin(4x^4-2)

Integral of 8x^3sin(4x^4+2) dx

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |     3    /   4    \   
 |  8*x *sin\4*x  + 2/ dx
 |                       
/                        
0

$$\int\limits_{0}^{1} 8 x^{3} \sin{\left(4 x^{4} + 2 \right)}\, dx$$

Integral((8*x^3)*sin(4*x^4 + 2), (x, 0, 1))

Detail solution

Let $u = 4 x^{4} + 2$ .

Then let $du = 16 x^{3} dx$ 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:
  
  $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
Now substitute $u$ back in:

$- \frac{\cos{\left(4 x^{4} + 2 \right)}}{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int 8 x^{3} \sin{\left(4 x^{4} + 2 \right)}\, dx = C - \frac{\cos{\left(4 x^{4} + 2 \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

cos(2)   cos(6)
------ - ------
  2        2

$$- \frac{\cos{\left(6 \right)}}{2} + \frac{\cos{\left(2 \right)}}{2}$$

cos(2)   cos(6)
------ - ------
  2        2

$$- \frac{\cos{\left(6 \right)}}{2} + \frac{\cos{\left(2 \right)}}{2}$$

cos(2)/2 - cos(6)/2

Numerical answer [src]

-0.688158561598754

-0.688158561598754

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

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Integral of 8x^3sin(4x^4+2) dx

Limits of integration:

The graph:

Piecewise:

The solution