If `int(e^(4x)-1)/(e^(2x))log((e^(2x)+1)/(e^(2x)-1))dx=(t^(2))/(2)logt-(t^(2))/(4)-(u^(2))/(2)logu+(u^(2))/(4)+C,` then

Options

A. (a) `u=e^(x)+e^(-x)`
B. (b) `u=e^(x)-e^(-x)`
C. (c) `t=e^(x)+e^(-x)`
D. (d) `t=e^(x)-e^(-x)`

Explanation

To solve the given integral equation, we need to evaluate the left-hand side (LHS) and compare it with the right-hand side (RHS) to find the values of \( u \) and \( t \).

### Step-by-Step Solution:

1. **Start with the LHS:**
\[ \int \frac{e^{4x} - 1}{e^{2x} \log\left(\frac{e^{2x} + 1}{e^{2x} - 1}\right)} \, dx \]

2. **Simplify the integrand:**
We can rewrite the integrand:
\[ \frac{e^{4x} - 1}{e^{2x}} = e^{2x} - e^{-2x} \]
Thus, the integral becomes:
\[ \int (e^{2x} - e^{-2x}) \log\left(\frac{e^{2x} + 1}{e^{2x} - 1}\right) \, dx \]

3. **Use properties of logarithms:**
The logarithm can be separated:
\[ \log\left(\frac{e^{2x} + 1}{e^{2x} - 1}\right) = \log(e^{2x} + 1) - \log(e^{2x} - 1) \]

4. **Rewrite the integral:**
The integral can now be expressed as:
\[ \int (e^{2x} - e^{-2x}) \left(\log(e^{2x} + 1) - \log(e^{2x} - 1)\right) \, dx \]

5. **Substitutions:**
Let:
\[ a = e^{x} + e^{-x}, \quad b = e^{x} - e^{-x} \]
Then, we differentiate:
\[ da = (e^{x} - e^{-x}) \, dx \quad \text{and} \quad db = (e^{x} + e^{-x}) \, dx \]

6. **Express the integral in terms of \( a \) and \( b \):**
The integral can be transformed into:
\[ \int a \log a \, da - \int b \log b \, db \]

7. **Integrate using integration by parts:**
For \( \int a \log a \, da \):
\[ = \frac{a^2}{2} \log a - \frac{a^2}{4} \]
For \( \int b \log b \, db \):
\[ = \frac{b^2}{2} \log b - \frac{b^2}{4} \]

8. **Combine the results:**
Thus, we have:
\[ \frac{a^2}{2} \log a - \frac{a^2}{4} - \left(\frac{b^2}{2} \log b - \frac{b^2}{4}\right) + C \]

9. **Compare with the RHS:**
The RHS is given as:
\[ \frac{t^2}{2} \log t - \frac{t^2}{4} - \frac{u^2}{2} \log u + \frac{u^2}{4} + C \]
By comparing coefficients, we identify:
\[ t = a = e^{x} + e^{-x}, \quad u = b = e^{x} - e^{-x} \]

### Final Values:
Thus, the values of \( t \) and \( u \) are:
\[ t = e^{x} + e^{-x}, \quad u = e^{x} - e^{-x} \]