Integral of Square Root of Quadratic Expression

Find the integral $\displaystyle \int \frac{\sqrt{t^2 + 2t}}{t + 1} \, dt$.

Integrating expressions like $\int \frac{\sqrt{t^2 + 2t}}{t + 1} \, dt$ often involves transforming the integrand into a more manageable form. One effective strategy in these scenarios is to simplify the expression inside the square root, possibly by completing the square for the quadratic expression. This can make subsequent techniques, like substitution, more straightforward. Additionally, identifying any algebraic manipulations that simplify the integrand can be crucial.

Often, problems involving integrals with square roots can also benefit from trigonometric or hyperbolic substitutions. Recognizing the form of the expression is key to selecting the appropriate substitution. In this case, completing the square simplifies the understanding of the structure, and can point towards a substitution that makes the integral solvable. Moreover, the choice of substitution or manipulation should aim to convert the integral into a standard form that is easier to evaluate.

This problem illustrates the importance of choosing a strategic approach when dealing with integrals involving square roots in the numerator and linear terms in the denominator. It's not just about applying rote techniques but about understanding the underlying structure and finding the most elegant path to the solution.

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