Inverse Trig Functions and Standard Integrals

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Inverse Trig Functions and Standard Integrals
The proof is on page 60 if you wish to read it.
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Page 61 Exercise 1A Questions 1(a), (b), (e),
(f), 2(a), (c), (f), (h). Page 62 Exercise 1B
Questions 1(a), (c), (e), (h)
TJ Exercise 1
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Integration of Rational Functions.
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Page 64 Exercise 2 Questions 5(a), (b), (d), (e),
6(a), (c) Page 66 Exercise 3A Questions 1 to 3
and 7
TJ Exercise 2, 3 and 4
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Integration By Parts
A method based on the product rule.
Sometimes we are asked to integrate the product
of two functions. To see how to do this, let us
examine the product rule for differentiation.
Integrating both sides with respect to x
Re-arranging gives
The aim is to make the new integral simpler than
the first.
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Example 1.
Let
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Example 2.
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Page 69 Exercise 4 Questions 1 and 2
TJ Exercise 5
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Developing Integration by Parts
Sometimes, the process of integrating by parts
must be applied more than once in order to solve
the integral.
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Let
Let
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2. Evaluate
Let
Let
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3. Evaluate
Note that the derivative of neither function is
simpler than the original function.
Let
Let
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Page 71 Exercise 5A Questions 1(a) to (g), (m),
(n), (q), (s).
TJ Exercise 6
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Integration by parts involving a Dummy Function
Functions like ln(x), sin-1 x, cos-1 x, and tan-1
x do not have a standard integral but have a
standard derivative. In order to integrate them,
we introduce a dummy function, namely the
number 1.
i.e let f (x) 1.
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Example 1.
Let
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Example 2.
Let
For
Use the substitution
i.e.
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TJ Exercise 7 and 8
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Differential Equations
If an equation contains a derivative then it is
called a differential equation.
A function which satisfies the equation is called
a solution to the differential equation.
The order of a differential equation is the order
of the highest derivative involved.
The degree of a differential equation is the
degree of the power of the highest derivative
involved.
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This section deals only with first order, first
degree differential equations.
Differential equations are solved by integration.
When the solution contains the constant of
integration it is called a general solution.
When we are given some initial conditions which
allow us to evaluate this constant the resultant
solution is called a particular solution.
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Integrating with respect to x.
Thus the particular solution is
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Integrating with respect to x.
Thus the particular solution is
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Further Differential Equations
We can now obtain the general solution.
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using the initial conditions,
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Variables separable
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using the initial conditions,
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Page 77 Exercise 8 Questions 1, 2, 3 and 5
TJ Exercise 9 and 10
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Applications of differential equations
Newtons law of cooling states that the rate at
which an object cools is proportional to the
difference between its temperature and that of
its surroundings.
Let T be the temperature difference at time t.
If T T0 at t 0, express T in terms of t.
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Page 81 Exercise 9A Questions 2 and 4 to 9
TJ Exercise 11
Do the review on page 86