Evaluate the integrals in Exercises 47–68.


∫₁² 4 dv
v²

In Exercises 75–78, find dy/dx.

y = ∫(from x to 1) (6/(3 + t^4))dt

Evaluate the integrals in Exercises 47–68.

∫₀¹/² x³ (1 + 9x⁴)⁻³/² dx

Definite Integrals

In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.

     n

 lim  ∑ (2cₖ - 1)⁻¹/² ∆xₖ, where P is a partition of [1, 5]

 ∥P∥→0  k = 1

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.

y = sin x, y = x, 0 ≤ x ≤ π/4

Express the solutions of the initial value problems in Exercises 35 and 36 in terms of integrals.

dy/dx = sin x/x , y(5) = -3

If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.

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d. ∫₀² √2ƒ(x) dx