Evaluate integrals

Please see the attached file for the fully formatted problems. Evaluate the following integrals. S (4x^3 -2x - (2/x^3) dx S (1/2x^1/2) dx 1-->0 S ln x dx

Rectilinear motion: Calculation of velocity

A billiard ball is hit and travels in a line. If s centimeters is the distance of the ball from its initial position at t seconds, then s=100t2 + 100t. If the ball hits a cushion that is 39cm from its initial position, at what velocity does it hit the cushion?

Finding a tangent line to a curve.

Find an equation of the tangent line to the curve, Y = x^3 - 3x^2 + 5x that has the least slope.

Differentiation: Shadow Shortening Problem

A man 6 ft tall is walking toward a building at the rate of 5 ft/sec. If there is a light on the ground 50 ft from the building, how fast is the man's shadow on the building growing shorter when he is 30ft from the building?

Finding the equation of a tangent to a curve that has the least slope.

Find an equation of the tangent line to the curve Y= x3 - 3x2 + 5x that has the least slope.

Finding the tangent to a curve. Picture included.

Find an equation of the tangent line to the curve Y= x3 - 3x2 + 5x that has the least slope.

Derivatives from first principles using standard theorems on trigonometric limits.

Using the definition of the derivative and any standard limiting theorems, show that the derivative of (sinx)^2 is sin(2x).

Implicit differentiation : the chain rule and product rule.

Use implicit differentiation to find dy/dx if y^2 + 3xy + x^2 + 10 = 0 (1) where y is a function of the independent variable x.

Finding the derivative of the tangent function by an application of the quotient rule.

Using the quotient rule, obtain an expression for the derivative of tan(x). State the values for which this is well-defined.

Maclaurin Series

I am trying to find the Maclaurin series for the function f(x) = sin2x/3x.