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Integral of d{x}:
Integral of cosx
Integral of x^e
Integral of ×^2
Integral of 1/(x³+1)
Identical expressions
x^ two *exp(three *x)
x squared multiply by exponent of (3 multiply by x)
x to the power of two multiply by exponent of (three multiply by x)
x2*exp(3*x)
x2*exp3*x
x²*exp(3*x)
x to the power of 2*exp(3*x)
x^2exp(3x)
x2exp(3x)
x2exp3x
x^2exp3x
x^2*exp(3*x)dx

Solving integrals/ x^2*exp(3*x)

Integral of x^2exp(3x) dx

The solution

You have entered [src]

  1           
  /           
 |            
 |   2  3*x   
 |  x *e    dx
 |            
/             
0

$$\int\limits_{0}^{1} x^{2} e^{3 x}\, dx$$

Integral(x^2*exp(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 $\operatorname{dv}{\left(x \right)} = e^{3 x}$ .

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 $\frac{du}{3}$ :
  
  $\int \frac{e^{u}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{3}$
  Now substitute $u$ back in:
  
  $\frac{e^{3 x}}{3}$
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 $\operatorname{dv}{\left(x \right)} = e^{3 x}$ .

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 $\frac{du}{3}$ :
  
  $\int \frac{e^{u}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{3}$
  Now substitute $u$ back in:
  
  $\frac{e^{3 x}}{3}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{2 e^{3 x}}{9}\, dx = \frac{2 \int e^{3 x}\, dx}{9}$
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{e^{u}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{3}$
  Now substitute $u$ back in:
  
  $\frac{e^{3 x}}{3}$
So, the result is: $\frac{2 e^{3 x}}{27}$
Now simplify:

$\frac{\left(9 x^{2} - 6 x + 2\right) e^{3 x}}{27}$
Add the constant of integration:

$\frac{\left(9 x^{2} - 6 x + 2\right) e^{3 x}}{27}+ \mathrm{constant}$

The answer is:

$\frac{\left(9 x^{2} - 6 x + 2\right) e^{3 x}}{27}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                            
 |                     3*x        3*x    2  3*x
 |  2  3*x          2*e      2*x*e      x *e   
 | x *e    dx = C + ------ - -------- + -------
 |                    27        9          3   
/

$$\int x^{2} e^{3 x}\, dx = C + \frac{x^{2} e^{3 x}}{3} - \frac{2 x e^{3 x}}{9} + \frac{2 e^{3 x}}{27}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

          3
  2    5*e 
- -- + ----
  27    27

$$- \frac{2}{27} + \frac{5 e^{3}}{27}$$

          3
  2    5*e 
- -- + ----
  27    27

$$- \frac{2}{27} + \frac{5 e^{3}}{27}$$

-2/27 + 5*exp(3)/27

Numerical answer [src]

3.64546980059031

3.64546980059031

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of x^2exp(3x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Integral of x^2*exp(3*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of x^2exp(3x) dx