in the integral limit of a to 0 of 4x/9+4x squared dx =1/2ln5  determine the value of  a

To solve the integral:

\[
\int_a^0 \frac{4x}{9 + 4x^2} \, dx = \frac{1}{2} \ln 5
\]

We need to evaluate this definite integral and find the value of \( a \). Let's proceed step by step.

### Step 1: Substitution
Let’s use a substitution to simplify the integral. Set:

\[
u = 9 + 4x^2
\]

Then, the derivative of \( u \) with respect to \( x \) is:

\[
\frac{du}{dx} = 8x \quad \text{or} \quad du = 8x \, dx
\]

Thus, \( \frac{du}{2} = 4x \, dx \), which matches part of the original integral.

### Step 2: Adjust the Limits of Integration
When \( x = a \), we have:

\[
u = 9 + 4a^2
\]

When \( x = 0 \), we have:

\[
u = 9
\]

So, the limits of integration for \( u \) will change from \( 9 + 4a^2 \) to \( 9 \).

### Step 3: Substituting into the Integral
Substitute into the original integral:

\[
\int_a^0 \frac{4x}{9 + 4x^2} \, dx = \int_{9 + 4a^2}^{9} \frac{du}{2u}
\]

This simplifies to:

\[
\frac{1}{2} \int_{9 + 4a^2}^{9} \frac{du}{u}
\]

### Step 4: Evaluate the Integral
The integral of \( \frac{1}{u} \) is \( \ln|u| \), so we have:

\[
\frac{1}{2} \left[ \ln u \right]_{9 + 4a^2}^{9}
\]

This becomes:

\[
\frac{1}{2} \left( \ln 9 - \ln(9 + 4a^2) \right)
\]

Using the properties of logarithms, we simplify this to:

\[
\frac{1}{2} \ln \left( \frac{9}{9 + 4a^2} \right)
\]

### Step 5: Set the Integral Equal to \( \frac{1}{2} \ln 5 \)
We are given that this expression equals \( \frac{1}{2} \ln 5 \), so:

\[
\frac{1}{2} \ln \left( \frac{9}{9 + 4a^2} \right) = \frac{1}{2} \ln 5
\]

Cancel the \( \frac{1}{2} \) on both sides:

\[
\ln \left( \frac{9}{9 + 4a^2} \right) = \ln 5
\]

### Step 6: Exponentiate Both Sides
Exponentiating both sides to eliminate the logarithms:

\[
\frac{9}{9 + 4a^2} = 5
\]

### Step 7: Solve for \( a \)
Now, solve for \( a \):

\[
9 = 5(9 + 4a^2)
\]

\[
9 = 45 + 20a^2
\]

\[
-36 = 20a^2
\]

\[
a^2 = \frac{-36}{20} = -\frac{9}{5}
\]

Since this result gives a negative value for \( a^2 \), it appears there is no real solution for \( a \). This suggests that the setup or the interpretation of the integral may need to be revisited for complex numbers, as \( a^2 \) being negative leads to imaginary numbers.

Would you like to explore further with complex numbers or recheck any assumptions?

### Related Questions:
1. How do we interpret an integral with imaginary bounds?
2. What substitution techniques simplify integrals involving \( x^2 \)?
3. How do logarithmic properties help in solving integrals?
4. What other methods can solve rational integrals like this one?
5. 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.

In the integral limit of a to 0 of 4x/(9 + 4x^2) dx = 1/2 ln 5, determine the value of a.

Explore how to solve the integral ∫ 4x/(9 + 4x²) dx with a detailed substitution method, logarithmic functions, and step-by-step evaluation. Determine the value of 'a' where the integral equals 1/2 ln 5. Suitable for undergraduate calculus students.

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