Integrate x^4 / (1 - x^2)^(3/2)

Thread starter mat331760298
Start date Jan 29, 2011
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Integrate

In summary, the formula for integrating x^4 / (1 - x^2)^(3/2) is ∫(x^4)/(1 - x^2)^(3/2) dx = -(x^3)/√(1 - x^2) + (3/2)∫(x^2)/(1 - x^2)^(1/2) dx. The method for solving this integral is by using substitution, specifically letting u = 1 - x^2 and du = -2x dx. The limits of integration for this integral depend on the given problem. Unfortunately, there is no shortcut or simpler method for solving this integral. Some real-life applications of this integral include calculating

Jan 29, 2011

mat331760298

1. Integral of (x^4)/(1-x^2)^(3/2) dx

i let x=sint so dx=costdt used the right triangle and simplified to integral of ((sint)^4)/((cost)^2) dt

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Jan 29, 2011

rock.freak667

Homework Helper

Now rewrite sin⁴t as (1-cos²t)² and expand.

What is the formula for integrating x^4 / (1 - x^2)^(3/2)?

The formula for integrating x^4 / (1 - x^2)^(3/2) is ∫(x^4)/(1 - x^2)^(3/2) dx = -(x^3)/√(1 - x^2) + (3/2)∫(x^2)/(1 - x^2)^(1/2) dx

What is the method for solving this integral?

The method for solving this integral is by using substitution. Let u = 1 - x^2 and du = -2x dx. This will result in the integral becoming -(1/2)∫u^(-3/2) du which can be solved using the power rule.

What are the limits of integration for this integral?

The limits of integration for this integral depend on the original problem. If the integral is given in the form ∫a^b (x^4)/(1 - x^2)^(3/2) dx, then the limits of integration would be a and b. If the limits are not given, then they would need to be determined from the problem's context.

Is there a shortcut or simpler method for solving this integral?

Unfortunately, there is no shortcut or simpler method for solving this integral. It requires the use of substitution and the power rule to solve it.

What are some real-life applications of this integral?

This integral can be used in physics and engineering to calculate the moment of inertia for a circular disc or hoop. It can also be used in calculating the electric field due to a charged ring or disk. Additionally, it can be used in economics to calculate consumer surplus in a market with a perfect price discrimination model.