7–64. Integration review Evaluate the following integrals.
49. ∫ √(9 + √(t + 1)) dt

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Step 1: Begin by analyzing the integral ∫ √(9 + √(t + 1)) dt. Notice that the integrand involves nested square roots, which suggests substitution might simplify the expression.

Step 2: Let u = √(t + 1). Then, differentiate u with respect to t: du/dt = 1/(2√(t + 1)), or equivalently, dt = 2u du.

Step 3: Substitute u = √(t + 1) and dt = 2u du into the integral. The integral becomes ∫ √(9 + u) * 2u du.

Step 4: Simplify the integral further by factoring out constants and rewriting the expression. The integral now becomes 2 ∫ u√(9 + u) du.

Step 5: To evaluate this integral, consider using another substitution or expanding the integrand. For example, let v = 9 + u, which simplifies √(9 + u) and allows further integration steps. Proceed with substitution and integration techniques to solve.

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Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and trigonometric identities. Understanding these techniques is essential for simplifying complex integrals into more manageable forms. In this case, recognizing the structure of the integrand can guide the choice of an appropriate technique.

Substitution Method

The substitution method is a powerful technique in integration that involves changing the variable of integration to simplify the integral. By substituting a new variable for a function within the integral, one can often transform a complicated integral into a simpler one. This method is particularly useful when dealing with nested functions, as seen in the integral provided.

Definite vs. Indefinite Integrals

Definite integrals compute the area under a curve between two specified limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is crucial for correctly interpreting the results of an integral. In this question, the integral is indefinite, meaning the result will include an arbitrary constant.

7–64. Integration review Evaluate the following integrals.49. ∫ √(9 + √(t + 1)) dt