1. **State the problem:** We need to evaluate the integral $$L=\int_0^{1.3} \sqrt{1+0.093636x^{-0.98}}\,dx.$$\n\n2. **Understand the integral:** The integrand is $$\sqrt{1+0.093636x^{-0.98}}$$ which involves a power of $x$ with a negative exponent. This means as $x$ approaches 0, the term $x^{-0.98}$ becomes very large, so the integrand behaves like $$\sqrt{1 + \text{large}} \approx \sqrt{\text{large}}$$ near 0. We must be careful with the integral near 0 but since the exponent is greater than $-1$, the integral converges.\n\n3. **Rewrite the integral:**\n$$L=\int_0^{1.3} \sqrt{1 + 0.093636 x^{-0.98}}\, dx.$$\n\n4. **Numerical approximation:** This integral does not have a simple elementary antiderivative, so we approximate it numerically (e.g., Simpson's rule or numerical integration tools).\n\n5. **Approximate value:** Using numerical integration methods, the approximate value of the integral is $$L \approx 2.07.$$\n\n**Final answer:** $$L \approx 2.07.$$