Integral solved with partial fractions

Please see the attached file for the fully formatted problems. The integral is to be solved using partial fractions.

Sum of a series

I am stuck on how to solve the sum of the series that I have attached in a word document.

Calculating integrals: Sines and Cosines

Integrate {cos(x)cos(3x)cos(5x),x}

Simple Vector Cross Product Proof

Create a proof to show that the following is true. a x (b+c) = a x b + a x c

For the curve f(x) = x - 1/3x^2 (one third x squared), find the equation of the straight line which is tangent to this curve at the point x = 1.

For the curve f(x) = x - 1/3x^2 (one third x squared), find the equation of the straight line which is tangent to this curve at the point x = 1. See attachment for diagram.

Derivatives: Chain Rule

Please see the attached file for the fully formatted problems. For a composite function f(x) = g(u(x)) state the chain rule for the derivative f(x). For each of the following functions, compute the derivative, simplifying your answers. f(x)=ln(1 + x^2) f(x)= sin(x^2) f(x) = (sin x)^2

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?