Can't evaluate the Integral

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In summary, an integral that can't be evaluated means that there is no known method or formula for finding its exact value. However, there are various numerical methods, such as the trapezoidal rule and Monte Carlo integration, that can be used to approximate its value. Integration by parts and substitution can only be applied to well-behaved functions with a closed-form solution. It is possible for an integral to have no solution, known as being divergent, if the function is discontinuous or undefined. In addition, there are advanced techniques such as series expansions and contour integration that can be used to evaluate integrals that cannot be solved analytically.
  • #1
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Homework Statement


∫x2(√2+x)

Homework Equations


∫f(x) from a to b = f'(b) - f'(a)
and substitution rule

The Attempt at a Solution


[/B]
I decided to make u=√(2+x), du= 1/2√(2+x) and when solving dx, i got dx= 2√(2+x) du. Substituting and then simplifying, I managed to get ∫2(x^2)u^2 du. But i can't go further from there. i can't find a way to get rid of the x2, I didn't get far using u= x2 or u=x+2 so I'm pretty sure I'm using the right substitution. I need help.
 
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  • #2
romeIAM said:

Homework Statement


∫x2(√2+x)

Homework Equations


∫f(x) from a to b = f'(b) - f'(a)
and substitution rule

The Attempt at a Solution


[/B]
I decided to make u=√(2+x), du= 1/2√(2+x) and when solving dx, i got dx= 2√(2+x) du. Substituting and then simplifying, I managed to get ∫2(x^2)u^2 du. But i can't go further from there. i can't find a way to get rid of the x2, I didn't get far using u= x2 or u=x+2 so I'm pretty sure I'm using the right substitution. I need help.

Try making u=2+x. Then x=u-2. So x^2=(u-2)^2. Take it from there.
 
  • #3
romeIAM said:
I managed to get ∫2(x^2)u^2 du.
When you do a substitution, do a complete substitution. In this case, neither x nor dx should appear after you make the substitution.

Also, in your original integral, you omitted dx. It's not a good habit to get into to ignore the differential. Doing so will come back to bite you in other integration techniques, including trig substitution and integration by parts.
 
  • #4
romeIAM said:

Homework Statement


∫x2(√2+x)

The Attempt at a Solution


[/B]
I decided to make u=√(2+x), du= 1/2√(2+x) here. i can't find a way to get rid of the x2, I didn't get far using u= x2 or u=x+2 so I'm pretty sure I'm using the right substitution. I need help.

No need to substitute. Just expand the integrand, and integrate the sum by terms.
 

1. What does it mean when an integral can't be evaluated?

When an integral can't be evaluated, it means that there is no known method or formula for finding the exact value of the integral. This could be due to various reasons, such as the function being too complex or not having a closed-form solution.

2. Is there a way to approximate the value of an integral that can't be evaluated?

Yes, there are various numerical methods for approximating the value of an integral that can't be evaluated. Some common methods include the trapezoidal rule, Simpson's rule, and Monte Carlo integration. These methods involve dividing the interval of integration into smaller segments and using the function values at specific points to estimate the area under the curve.

3. Can't we use integration by parts or substitution to solve an integral that can't be evaluated?

Integration by parts and substitution are techniques used to evaluate definite integrals. However, they can only be applied if the integrand is a well-behaved function with a closed-form solution. If the function is too complex or doesn't have a known solution, these techniques cannot be used.

4. Is it possible for an integral to have no solution?

Yes, it is possible for an integral to have no solution. This can happen when the function being integrated is discontinuous or undefined at certain points, making it impossible to find a definite value for the integral. In such cases, the integral is said to be divergent.

5. Are there any alternative methods for evaluating an integral that can't be solved analytically?

Apart from numerical methods, there are also techniques such as series expansions, contour integration, and Laplace transforms that can be used to evaluate integrals that cannot be solved analytically. These methods are often used in more advanced areas of mathematics and science, such as engineering and physics.

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