1. **State the problem:** We want to find the integral $$L(t) = \int \frac{1+t}{t^2 + 3t} \, dt.$$ 2. **Rewrite the integrand:** Factor the denominator: $$t^2 + 3t = t(t+3).$$ So the integral becomes $$\int \frac{1+t}{t(t+3)} \, dt.$$ 3. **Use partial fraction decomposition:** We express $$\frac{1+t}{t(t+3)} = \frac{A}{t} + \frac{B}{t+3}$$ for constants $A$ and $B$. Multiply both sides by $t(t+3)$: $$1 + t = A(t+3) + Bt = At + 3A + Bt = (A + B)t + 3A.$$ 4. **Equate coefficients:** For $t$ terms: $1 = A + B$. For constant terms: $1 = 3A$. 5. **Solve for $A$ and $B$:** From $1 = 3A$, we get $$A = \frac{1}{3}.$$ From $1 = A + B$, we get $$B = 1 - \frac{1}{3} = \frac{2}{3}.$$ 6. **Rewrite the integral:** $$L(t) = \int \left( \frac{1/3}{t} + \frac{2/3}{t+3} \right) dt = \frac{1}{3} \int \frac{1}{t} dt + \frac{2}{3} \int \frac{1}{t+3} dt.$$ 7. **Integrate:** $$L(t) = \frac{1}{3} \ln|t| + \frac{2}{3} \ln|t+3| + C.$$ 8. **Combine logarithms:** $$L(t) = \ln \left| t^{1/3} (t+3)^{2/3} \right| + C.$$ 9. **Rewrite in terms of $t^2 + 3t$:** Note that $$t^2 + 3t = t(t+3).$$ So $$\ln|t^2 + 3t| = \ln|t| + \ln|t+3|.$$ 10. **Express $L(t)$ as a combination of logarithms:** We want to write $$L(t) = \frac{1}{2} \ln|t^2 + 3t| + k \ln \left| \frac{t+3}{t} \right| + C$$ for some constant $k$. 11. **Find $k$ by matching coefficients:** From step 7, $$L(t) = \frac{1}{3} \ln|t| + \frac{2}{3} \ln|t+3| + C.$$ From the form, $$\frac{1}{2} \ln|t^2 + 3t| + k \ln \left| \frac{t+3}{t} \right| = \frac{1}{2} (\ln|t| + \ln|t+3|) + k (\ln|t+3| - \ln|t|) = \left( \frac{1}{2} - k \right) \ln|t| + \left( \frac{1}{2} + k \right) \ln|t+3|.$$ Set equal to the coefficients from step 7: $$\frac{1}{3} = \frac{1}{2} - k,$$ $$\frac{2}{3} = \frac{1}{2} + k.$$ 12. **Solve for $k$:** From the first equation: $$k = \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}.$$ Check with second equation: $$\frac{2}{3} = \frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3},$$ which is correct. 13. **Final answer:** $$L(t) = \frac{1}{2} \ln|t^2 + 3t| + \frac{1}{6} \ln \left| \frac{t+3}{t} \right| + C.$$ This matches option (b).