Integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)]

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In summary, the formula for integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)] is ∫c1*x^c2*ln[-1+sqrt(1+c3*x^c4)] dx. The term c1 represents a constant coefficient, x^c2 represents the variable raised to a power, ln[-1+sqrt(1+c3*x^c4)] represents the natural logarithm of the function, and dx represents the differential of the variable of integration. The process for integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)] involves using integration by parts, where one term is chosen as the u-sub
  • #1
natlight
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Hi,

Can anyone suggest how to integrate the following function, in which all the c's are constants:

c1 * x^c2 * ln [ -1 + sqrt (1 + c3 * x^c4) ]

Much obliged!
 
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  • #2
Well, Mathematica yields:

-(1/(2 (1 + c2)^2))
c1 x^(1 +
c2) (c4 +
c4 Hypergeometric2F1[1/2, (1 + c2)/c4, (1 + c2 + c4)/
c4, -c3 x^c4] - 2 (1 + c2) Log[-1 + Sqrt[1 + c3 x^c4]])

I am not familiar with the Hypergeometric2F1 function.
 

Related to Integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)]

1. What is the formula for "Integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)]"?

The formula for integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)] is ∫c1*x^c2*ln[-1+sqrt(1+c3*x^c4)] dx.

2. What is the meaning of each term in the formula for "Integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)]"?

The term c1 represents a constant coefficient, x^c2 represents the variable raised to a power, ln[-1+sqrt(1+c3*x^c4)] represents the natural logarithm of the function, and dx represents the differential of the variable of integration.

3. What is the process for integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)]?

The process for integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)] involves using integration by parts, where one term is chosen as the u-substitution and the other term is chosen as the differential. Then, the formula for integration by parts is applied: ∫u*dv = u*v - ∫v*du. This process is repeated until the integral can be solved.

4. Are there any special cases for integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)]?

Yes, there may be special cases where the integral cannot be solved using traditional methods. In these cases, numerical methods or computer software may be used to approximate the solution.

5. What are some practical applications of integrating c1*x^c2*ln[-1+sqrt(1+c3*x^c4)]?

This type of integral may arise in physical or mathematical models involving logarithmic functions, such as in population growth or radioactive decay. It may also be used in engineering or physics problems involving natural logarithms. Additionally, it can be used in calculating areas under curves in calculus.

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