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A spring with a spring constant k is connected to the ceiling. First a weight of mass m is connected to the spring. Deduce the new equilibrium position of the spring, find its equation of motion and hence deduce its frequency f.

Let's first find the equilibrium position of the spring. When the mass is first attached the spring will oscillate like a simple harmonic oscillator, in the real world the oscillator will eventually settle a...
BP
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If the complex number z = 5 + 4i, work out 1/z.

1/(5 + 4i) * (5 - 4i)/(5 - 4i) = (5 - 4i)/41
OG
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How to integrate ln(x)?

You need to use a clever trick for this! Write ln(x) as 1 ln(x), and use integration by parts:u=ln(x) v'=1u'=1/x v=xThen applying the formula we obtain∫ln(x)dx = x ln(x) - ∫[(1/x) x] dx = = x ln(x) - ∫1 dx =...
KW
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Differentiate arcsin(2x) using the fact that 2x=sin(y)

Differentiate implicitly on both sides with respect to x to get: 2=cos(y) • (dy/dx). Divide by cos(y) on both sides to get: dy/dx=2/cos(y). Use the trigonometric identity cos^2(y)+sin^2(y)=1 rearranged to co...
LO
3652 Views

f(x)=ln(x). Find the area underneath the curve f(x) between 1 and 2.

We cannot dirrectly intergrate ln(x), so instead we intergrate 1 ln(x) using intergration by parts. The formula for intergration by parts is: ∫ (u dv/dx) dx = uv − ∫ v du/dx dx . We let u=ln(x) so that du/dx...
AT
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