Factor ##\ 3x^2-8x+5\ ##. Then putting your u substitution into that simplifies things a bit.gruba said:Homework Statement
Find the integral [itex]\int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx[/itex]
2. The attempt at a solution
I can't find a useful substitution to solve this integral.
I tried [tex]x-2=\frac{1}{u},x=\frac{1}{u}+2,dx=-\frac{1}{u^2}du[/tex] that gives
[tex]\int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx=-\int \frac{u}{\sqrt{3\left(\frac{1}{u}+2\right)^2-8\left(\frac{1}{u}+2\right)+5}}\mathrm du[/tex]
Is there a useful substitution for [itex]u[/itex]?
An irrational function is a type of mathematical function that contains irrational numbers, which cannot be expressed as a ratio of two integers. These functions involve variables raised to irrational powers, such as square roots or pi.
The process of integrating an irrational function involves using various techniques, such as substitution, integration by parts, and partial fractions. It also requires a good understanding of algebra and the properties of irrational numbers.
Some common examples of irrational functions include square root functions, logarithmic functions, and trigonometric functions such as sine and cosine. These functions often appear in real-world applications, such as in physics and engineering.
Integrating irrational functions allows us to solve complex mathematical problems and model real-world phenomena. It also helps us understand the behavior of these functions and their relationship to other types of functions.
One tip for solving integration problems involving irrational functions is to carefully identify which technique is most appropriate for the given function. It is also helpful to review the properties of irrational numbers and practice with a variety of problems to improve problem-solving skills.