Evaluate each of the following integrals. a) integral_1^3 (x^2 + 4x + 3/x^2) dx.

Question

Evaluate each of the following integrals. a) integral_1^3 (x^2 + 4x + 3/x^2) dx. b) integral_0^1 (x^1/2 - 2x + 1) dx. c) integral_0^2 (2/x + 3e^x) dx. d) integral_0^pi(2x - cos(x)) dx. e) integral(1/Squareroot x + sin h(x)) dx. Find the solution of the initial value problem dy/dx = 6x^2 + 4x where y(2) = 10. Find the area of the region bounded by f(x) = x(x - pi) and g(x) = sin(x). Find the equation of the tangent line to the curve x = t^3 + 3t, y = t^2 + 2t for t = 1. Approximate integral_0^1 In(x + 1) dx by subdividing the interval into 4 subintervals of equal length and using the right-hand sum approximation. A stone is thrown upward from the top of a 320-foot cliff at an initial velocity of 128 ft/sec eventually falls to the beach below. a) How long does it take the stone to reach its highest point? b) What is the maximum height? c) How long before the stone hits the beach? d) What is the velocity of the stone on impact? Let F(x) = integral_-1^x Squareroot 3t^2 + 1 dt, for x > -1. Find F'(x). Where is F(x) increasing and decreasing? Where is F(x) concave up and concave down? Graph F.

Explanation / Answer

3)given f(x)=x(x-), g(x)=sinx

x(x-)=sinx

=>x =0 ,

area = [0 to ]sinx -x(x-) dx

area = [0 to ]sinx -x2+x dx

area = [0 to ]-cosx -(1/3)x3+(/2)x2

area = [-cos -(1/3)3+(/2)2 ]-[-cos0 -(1/3)03+(/2)02 ]

area = 1 -(3/3)+(3/2) +1 +0-0

area = 2 +(3/6)

area =7.17 square units

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