How can I solve this tricky exponent integral problem using proper methods?

In summary, the conversation discusses solving an integral using various methods. The person got stuck with the integral \int x^2 e^{\frac{1}{2}x^2+x} dx and couldn't solve it with any method until combining them to get the answer \int x^2 e^{\frac{1}{2}x^2+x} dx=C+(x-1)e^{\frac{1}{2}x^2+x}. They are seeking advice on how to solve it properly, and one method suggested is using part integration. The conversation ends with the person thanking the expert for their clever solution and realizing the beauty of it.
  • #1
vabamyyr
66
0
i solved ODE with Lagrange method and got stuck with integral

[tex]\int x^2 e^{\frac{1}{2}x^2+x} dx[/tex]

i couldn't solve it with any method but combined and got the answer that it is

[tex]\int x^2 e^{\frac{1}{2}x^2+x} dx=C+(x-1)e^{\frac{1}{2}x^2+x}[/tex]

the problem is that i want to do it with proper method, and show how it comes out. My mentor said that these types of integrals are "freaky" with little twist:smile:

Anyway, i don't mind some advice
 
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  • #2
U can do it using part integration.The derivative of the exponent is [itex] x+1[/itex] which u can obtain writing [itex] x^{2}=x(x+1)-x [/itex]


Daniel.
 
  • #3
dextercioby said:
U can do it using part integration.The derivative of the exponent is [itex] x+1[/itex] which u can obtain writing [itex] x^{2}=x(x+1)-x [/itex]


Daniel.
i don't get how that helps me
 
  • #4
It does

[tex] \int x^{2}e^{\frac{1}{2}x^{2}+x} \ dx=\int \left(x^{2}+x-x\right) e^{\frac{1}{2}x^{2}+x} \ dx=\int x(x+1)e^{\frac{1}{2}x^{2}+x} \ dx-\int x e^{\frac{1}{2}x^{2}+x} \ dx[/tex]
[tex]=xe^{\frac{1}{2}x^{2}+x}-\int e^{\frac{1}{2}x^{2}+x} \ dx-\int x e^{\frac{1}{2}x^{2}+x} \ dx [/tex]
[tex]=x e^{\frac{1}{2}x^{2}+x}-\int (x+1)e^{\frac{1}{2}x^{2}+x} \ dx=x e^{\frac{1}{2}x^{2}+x}-e^{\frac{1}{2}x^{2}+x}+C [/tex]


Daniel.
 
  • #5
wow, that is very clever, at first i looked ur answer and couldn`t get one line but then i realized that the key moment was to crack int x(x+1)*e^...dx with simple method by taking u=x and dv=(x+1)*e^...dx and now i realize the beauty. So thank u very much for helping me on this one.
 

1. What is an exponent integral problem?

An exponent integral problem is a mathematical problem that involves solving for the integral of a function that contains exponents. This can be a challenging problem because it requires using techniques such as integration by parts and substitution to solve.

2. How do you solve an exponent integral problem?

To solve an exponent integral problem, you need to use techniques such as integration by parts, substitution, and partial fraction decomposition. It is also important to have a good understanding of the properties of exponents and logarithms.

3. What are some common types of exponent integral problems?

Some common types of exponent integral problems include integrals with exponential functions, logarithmic functions, and trigonometric functions raised to an exponent. These problems can also involve combinations of these functions.

4. Why are exponent integral problems important?

Exponent integral problems are important because they have many real-world applications, especially in science and engineering. They also help to develop critical thinking and problem-solving skills.

5. What are some tips for solving exponent integral problems?

Some tips for solving exponent integral problems include practicing different techniques, breaking the problem into smaller parts, and being familiar with the properties of exponents and logarithms. It is also helpful to check your answer using differentiation to ensure it is correct.

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