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Solving integrals/ 4x^3*sin(x)

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

The solution

You have entered [src]

  2               
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 |     3          
 |  4*x *sin(x) dx
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1

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

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

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = 4 x^{3}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 12 x^{2}$ .

To find $v{\left(x \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = - 12 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - 24 x$ .

To find $v{\left(x \right)}$ :
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = - 24 x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = -24$ .

To find $v{\left(x \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int 24 \cos{\left(x \right)}\, dx = 24 \int \cos{\left(x \right)}\, dx$
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
So, the result is: $24 \sin{\left(x \right)}$
Add the constant of integration:

$- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}+ \mathrm{constant}$

The answer is:

$- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                         
 |                                                                          
 |    3                                3              2                     
 | 4*x *sin(x) dx = C - 24*sin(x) - 4*x *cos(x) + 12*x *sin(x) + 24*x*cos(x)
 |                                                                          
/

$$\int 4 x^{3} \sin{\left(x \right)}\, dx = C - 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-20*cos(1) + 12*sin(1) + 16*cos(2) + 24*sin(2)

$$- 20 \cos{\left(1 \right)} + 16 \cos{\left(2 \right)} + 12 \sin{\left(1 \right)} + 24 \sin{\left(2 \right)}$$

-20*cos(1) + 12*sin(1) + 16*cos(2) + 24*sin(2)

$$- 20 \cos{\left(1 \right)} + 16 \cos{\left(2 \right)} + 12 \sin{\left(1 \right)} + 24 \sin{\left(2 \right)}$$

-20*cos(1) + 12*sin(1) + 16*cos(2) + 24*sin(2)

Numerical answer [src]

14.456394559394

14.456394559394

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution