# Is it possible to solve this integral by hand or by conventional methods?

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#inte^(x^(2))*x^2dx#

I can use u sub, trig sub, integration by parts, tabular, or other conventional methods... I was told if this is not possible, just leave it be and use a calculator

I can use u sub, trig sub, integration by parts, tabular, or other conventional methods... I was told if this is not possible, just leave it be and use a calculator

##### 4 Answers

You can't indeed.

#### Explanation:

You could have solved

since it is written in the form

up to some constants. But the one you wrote has no solutions in terms of standard functions.

This can be solved using integration by parts and u-substitution being aware of a common integral form that results in the so-called "error function". If you are not familiar with the error function (

No, it is not.

#### Explanation:

Noting that:

integrate by parts:

Now the gaussian integral :

cannot be expressed in terms of elementary functions, so neither can the proposed function.

# int \ x^2e^(x^2) \ dx = sum_(n=0)^oo (x^(2n+3))/((2n+3)*n!) #

#### Explanation:

We seek:

# I = int \ x^2e^(x^2) \ dx #

There is no elementary solution in terms of traditional functions, however we can use:

# e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ... #

so that:

# I = int \ x^2 \ {1 + (x^2) + (x^2)^2/(2!) + (x^2)^3/(3!) + (x^2)^4/(4!) + ... } \ dx #

# \ \ = int \ x^2 \ {1 + x^2 + x^4/(2!) + x^6/(3!) + x^8/(4!) + ... } \ dx #

# \ \ = int \ x^2 + x^4 + x^6/(2!) + x^8/(3!) + x^10/(4!) + ... \ dx #

# \ \ = x^3/3 + x^5/5 + x^7/(7*2!) + x^9/(9*3!) + x^11/(11*4!) + ... \ dx #

# \ \ = sum_(n=0)^oo (x^(2n+3))/((2n+3)*n!) #