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

Guida verificata passo dopo passo

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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