Definite integral with some unknown variables

• Mathematica
• shinobi20
In summary, when trying to evaluate an integral with unknown variables in Mathematica, it is important to consider possible singularities and convergence issues. One way to address this is by explicitly defining assumptions for the variables and passing them to the software.
shinobi20
TL;DR Summary
I am evaluating a definite integral with unknown variables in Mathematica.
I am trying to evaluate an integral with unknown variables ##a, b, c## in Mathematica, but I am not sure why it takes so long for it to give an output, so I just decided to cancel the running. The integral is given by,

##\int_0^1 dy \frac{ y^2 (1 - b^3 y^3)^{1/2} }{ (1 - a^4 c^2 y^4)^{1/2} }##

Depending on the values of ##a, b, c## you can expect singularities in the integrand that may make the integral improper or even non-convergent. So, if the problem allows it, you should probably tell the software explicitly about assumptions on ##a, b, c##.

For example, I would begin by defining ##d := a^4 c^2## and passing the assumption that ##d < 1##.

Last edited:
Dale
S.G. Janssens said:
Depending on the values of a,b,c you can expect singularities in the integrand that may make the integral improper or even non-convergent. So, if the problem allows it, you should probably tell the software explicitly about assumptions on a,b,c.

For example, I would begin by defining d:=a4c2 and passing the assumption that d<1.
This is my code

Code:
d=2;
z=1;
b=a/z;
h=c^2 a^2d;
Integrate[ ( y^d (1 - (b y)^(d+1))^1/2 )/(1 - h^2d y^2d)^1/2 , {y,0,1}, Assumptions -> {y>0, h<1} ]

It does not return anything.

1. What is a definite integral with unknown variables?

A definite integral with unknown variables is a mathematical expression that involves finding the area under a curve or between two curves, where one or more of the variables in the expression is unknown. It is typically used in calculus to solve problems involving changing quantities.

2. How do you solve a definite integral with unknown variables?

To solve a definite integral with unknown variables, you need to use integration techniques such as substitution or integration by parts. You may also need to use algebraic manipulation to rearrange the expression and simplify it before integrating.

3. Can you give an example of a definite integral with unknown variables?

One example of a definite integral with unknown variables is ∫(3x + y)dx, where both x and y are unknown variables. This can be solved using integration by parts, where u = 3x and dv = dx.

4. What are the applications of definite integrals with unknown variables?

Definite integrals with unknown variables have many applications in physics, engineering, economics, and other fields where quantities are constantly changing. They can be used to calculate displacement, velocity, acceleration, work, and many other important quantities.

5. Are there any special rules for solving definite integrals with unknown variables?

Yes, there are some special rules for solving definite integrals with unknown variables. For example, if the limits of integration involve the unknown variable, you may need to use substitution to solve the integral. Additionally, if the expression involves trigonometric functions, special trigonometric identities may need to be used.

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