Integral of x^2sin3xdx dx

The solution

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

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

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(3 x \right)}$ .

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

To find $v{\left(x \right)}$ :
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:
  
  $- \frac{\cos{\left(3 x \right)}}{3}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = - \frac{2 x}{3}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(3 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - \frac{2}{3}$ .

To find $v{\left(x \right)}$ :
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\cos{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(3 x \right)}}{3}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{2 \sin{\left(3 x \right)}}{9}\right)\, dx = - \frac{2 \int \sin{\left(3 x \right)}\, dx}{9}$
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: $\frac{2 \cos{\left(3 x \right)}}{27}$
Add the constant of integration:

$- \frac{x^{2} \cos{\left(3 x \right)}}{3} + \frac{2 x \sin{\left(3 x \right)}}{9} + \frac{2 \cos{\left(3 x \right)}}{27}+ \mathrm{constant}$

The answer is:

$- \frac{x^{2} \cos{\left(3 x \right)}}{3} + \frac{2 x \sin{\left(3 x \right)}}{9} + \frac{2 \cos{\left(3 x \right)}}{27}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                    2                        
 |  2                   2*cos(3*x)   x *cos(3*x)   2*x*sin(3*x)
 | x *sin(3*x) dx = C + ---------- - ----------- + ------------
 |                          27            3             9      
/

$$\int x^{2} \sin{\left(3 x \right)}\, dx = C - \frac{x^{2} \cos{\left(3 x \right)}}{3} + \frac{2 x \sin{\left(3 x \right)}}{9} + \frac{2 \cos{\left(3 x \right)}}{27}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  2    7*cos(3)   2*sin(3)
- -- - -------- + --------
  27      27         9

$$- \frac{2}{27} + \frac{2 \sin{\left(3 \right)}}{9} - \frac{7 \cos{\left(3 \right)}}{27}$$

  2    7*cos(3)   2*sin(3)
- -- - -------- + --------
  27      27         9

$$- \frac{2}{27} + \frac{2 \sin{\left(3 \right)}}{9} - \frac{7 \cos{\left(3 \right)}}{27}$$

-2/27 - 7*cos(3)/27 + 2*sin(3)/9

Numerical answer [src]

0.213950649057864

0.213950649057864

The graph

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

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Identical expressions

Integral of x^2sin3xdx dx

Limits of integration:

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Piecewise:

The solution