Integrating x/(ax^3 + bx -c ): Solutions & Steps Explained

In summary, the conversation is about integrating the expression x/(ax^3+bx-c) and using partial fractions to simplify it. One person is struggling with the second term of the partial fractions and asks for help integrating it, as the quadratic expression in the denominator does not have real roots. The other person suggests completing the square in the denominator and using substitutions to integrate the term.
  • #1
grey
70
0
hi,

can anyone tell me how the following will be integrated:
(where all letters except 'x' are constants)

x/(ax^3 + bx -c )

i tried to simplify the integral using partial fractions, but ended up with:

p/(x-q) + (rx +s)/(tx^2 + ux + v)
obviously, the first term is trivial, but how to integrate the second term. (the quadratic expression in the denominator does not have real roots)
 
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  • #2
(rx +s)/(tx^2 + ux + v)

If the numerator is a constant times the derivative of the denominator, great, make a substitution.

Otherwise, complete the square in the denominator ... surely every calculus textbook has this method?
 
  • #3
grey said:
hi,

can anyone tell me how the following will be integrated:
(where all letters except 'x' are constants)

x/(ax^3 + bx -c )

i tried to simplify the integral using partial fractions, but ended up with:

p/(x-q) + (rx +s)/(tx^2 + ux + v)
obviously, the first term is trivial, but how to integrate the second term. (the quadratic expression in the denominator does not have real roots)
If a quadratic expression such as [itex]tx^2+ ux+ v[/itex] cannot be factored, then you can complete the square to get something like [itex]t(x-a)^2+ b[/itex] where b is positive.
Then [itex](rx+ x)/(t(x-a)^2+ b)[/itex] can be written as [itex](r(x-a)+ ra+ a)/(t(x-a)^2+ b)[/itex].
The first term, [itex]r(x-a)dx/(t(x-a)^2+ b)[/itex] can be integrated with the substitution u= (x-a)^2 so that (1/2)du= (x-a)dt and the integrand becomes (r/(2t))du/u. The second term, (ra+a)dx/(t(x-a)^2+ b)= 1/(b(ra+a)) dx/((t/b)(x-a)^2+ 1) can be integrated with the substitution [itex]u= \sqrt{t}{b}(x-a)[/itex] and the fact that [itex]\int du/(u^2+1)= arctan(u)+ C[/itex].
 
  • #4
thanks guys!

definite help! thanks!
 

FAQ: Integrating x/(ax^3 + bx -c ): Solutions & Steps Explained

What is the purpose of integrating x/(ax^3 + bx - c)?

The purpose of integrating x/(ax^3 + bx - c) is to find the area under the curve of the function, which represents the accumulation of the function over a given interval. This is useful in many scientific fields, such as physics and engineering, where understanding the rate of change of a variable is important.

What are the general steps for integrating x/(ax^3 + bx - c)?

The general steps for integrating x/(ax^3 + bx - c) are:
1. Rewrite the function using partial fraction decomposition to break it into simpler fractions.
2. Apply the power rule to integrate each individual fraction.
3. Combine the results and add the constant of integration.
4. Check the validity of the solution by taking the derivative of the integrated function and comparing it to the original function.

How do you handle complex or non-standard integrals of x/(ax^3 + bx - c)?

If the integral of x/(ax^3 + bx - c) cannot be solved using the general steps, other techniques such as substitution, integration by parts, or trigonometric substitution may be necessary. In some cases, the integral may not have an analytic solution and numerical methods may be used to approximate the solution.

Can you provide an example of integrating x/(ax^3 + bx - c)?

For example, let's integrate x/(2x^3 + 3x - 5).
Step 1: Rewrite the function using partial fraction decomposition:
x/(2x^3 + 3x - 5) = x/(2x - 5)(x^2 + 2x + 1)
= A/(2x - 5) + (Bx + C)/(x^2 + 2x + 1)
Where A, B, and C are constants to be determined.
Step 2: Integrate each individual fraction:
∫ A/(2x - 5) dx = A/2 ln|2x - 5| + C
∫ (Bx + C)/(x^2 + 2x + 1) dx = B ln|x^2 + 2x + 1| - 2B arctan(x + 1) + C
Step 3: Combine the results and add the constant of integration:
∫ x/(2x^3 + 3x - 5) dx = A/2 ln|2x - 5| + B ln|x^2 + 2x + 1| + (-2B - A) arctan(x + 1) + C
Step 4: Check the validity of the solution by taking the derivative:
d/dx (A/2 ln|2x - 5| + B ln|x^2 + 2x + 1| + (-2B - A) arctan(x + 1) + C)
= A/(2x - 5) + (Bx + C)/(x^2 + 2x + 1)
= x/(2x^3 + 3x - 5)
Therefore, the solution is valid.
Note: The constants A, B, and C can be solved using either the method of undetermined coefficients or by equating coefficients of like terms.

How can integrating x/(ax^3 + bx - c) be applied in real-world scenarios?

Integrating x/(ax^3 + bx - c) has many applications in real-world scenarios. For example, in physics, it can be used to calculate the work done by a variable force over a certain distance. In economics, it can be used to calculate the total revenue from a demand curve. In engineering, it can be used to determine the deflection of a beam under a varying load. In general, it is a powerful tool for understanding the behavior of various systems and variables in real-world situations.

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