Math Problem Statement
in the integral limit of a to 0 of 4x/9+4x squared dx =1/2ln5 determine the value of a
Solution
To solve the integral:
We need to evaluate this definite integral and find the value of . Let's proceed step by step.
Step 1: Substitution
Let’s use a substitution to simplify the integral. Set:
Then, the derivative of with respect to is:
Thus, , which matches part of the original integral.
Step 2: Adjust the Limits of Integration
When , we have:
When , we have:
So, the limits of integration for will change from to .
Step 3: Substituting into the Integral
Substitute into the original integral:
This simplifies to:
Step 4: Evaluate the Integral
The integral of is , so we have:
This becomes:
Using the properties of logarithms, we simplify this to:
Step 5: Set the Integral Equal to
We are given that this expression equals , so:
Cancel the on both sides:
Step 6: Exponentiate Both Sides
Exponentiating both sides to eliminate the logarithms:
Step 7: Solve for
Now, solve for :
Since this result gives a negative value for , it appears there is no real solution for . This suggests that the setup or the interpretation of the integral may need to be revisited for complex numbers, as being negative leads to imaginary numbers.
Would you like to explore further with complex numbers or recheck any assumptions?
Related Questions:
- How do we interpret an integral with imaginary bounds?
- What substitution techniques simplify integrals involving ?
- How do logarithmic properties help in solving integrals?
- What other methods can solve rational integrals like this one?
- What conditions lead to real vs imaginary solutions in integration?
Tip:
When dealing with complex solutions, keep an eye on the discriminant in quadratic equations—it helps identify whether the solutions are real or imaginary.
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Math Problem Analysis
Mathematical Concepts
Integration
Substitution in Integrals
Logarithmic Functions
Formulas
Substitution: u = 9 + 4x^2
Logarithmic Identity: ln(a) - ln(b) = ln(a/b)
Definite Integral: ∫ du/u = ln|u|
Theorems
Fundamental Theorem of Calculus
Logarithmic Properties
Suitable Grade Level
Undergraduate Calculus
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