Solving a Complex Integral: Help Needed!

In summary, the conversation revolves around finding the integral of a rational expression involving t^3, t^2, and t. The suggested method is to use partial fractions decomposition to rewrite the expression as the sum of simpler rational expressions, which can then be integrated.
  • #1
skyturnred
118
0

Homework Statement



[itex]\int^{√x}_{1}[/itex][itex]\frac{t^{3}+t-1}{t^{2}(t^{2}+1)}[/itex] dt

Homework Equations





The Attempt at a Solution



So I first start by expanding the bottom part of the fraction to t[itex]^{4}[/itex]+t[itex]^{2}[/itex], and letting u equal to that. Then du=4t[itex]^{3}[/itex]+2t dt. I move the common multiple of 2 over to the other side so that it is (1/2)du=2t[itex]^{3}[/itex]+t dt. I cannot find out how to relate that to the numerator (although I am so close).

Can someone please help? Thanks!
 
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  • #2


skyturnred said:

Homework Statement



[itex]\int^{√x}_{1}[/itex][itex]\frac{t^{3}+t-1}{t^{2}(t^{2}+1)}[/itex] dt

Homework Equations





The Attempt at a Solution



So I first start by expanding the bottom part of the fraction to t[itex]^{4}[/itex]+t[itex]^{2}[/itex], and letting u equal to that. Then du=4t[itex]^{3}[/itex]+2t dt. I move the common multiple of 2 over to the other side so that it is (1/2)du=2t[itex]^{3}[/itex]+t dt. I cannot find out how to relate that to the numerator (although I am so close).

Can someone please help? Thanks!
Do you know partial fractions decomposition? That seems to me to be the way to go. Using that technique you rewrite (t3 + t - 1)/(t2(t2 + 1) as the sum of three rational expressions of the form
[tex]\frac{A}{t} + \frac{B}{t^2} + \frac{Ct + D}{t^2 + 1}[/tex]

The idea is to find constants A, B, C, D so that the new representation is identically equal to the original rational expression. Once you find the constants, then integrate the sum of simpler functions.
 

1. What is a complex integral?

A complex integral is a mathematical expression that involves the integration of a complex-valued function. It is similar to a regular integral, but the function being integrated is complex rather than real-valued.

2. How do you solve a complex integral?

Solving a complex integral involves using various techniques from complex analysis, such as contour integration, Cauchy's integral theorem, and the residue theorem. These techniques allow for the evaluation of complex integrals in a systematic way.

3. Why are complex integrals important?

Complex integrals are important in many areas of mathematics and physics, including complex analysis, differential equations, and quantum mechanics. They allow for the calculation of complex-valued quantities that cannot be obtained through regular integration methods.

4. Can complex integrals have multiple solutions?

Yes, complex integrals can have multiple solutions. This is because complex functions can have multiple branch cuts and branch points, leading to different solutions for the same integral depending on the chosen branch. It is important to specify the branch being used in order to obtain a unique solution.

5. What are some tips for solving complex integrals?

Some tips for solving complex integrals include choosing an appropriate contour, understanding the behavior of the integrand, and utilizing symmetry and other properties of the function. It is also helpful to have a good understanding of complex numbers and complex analysis techniques.

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