Advanced Higher Notes

Inverse Trigonometric Functions Integration By Partial Fractions 1 Integration By Partial Fractions 2 Integration By Partial Fractions 3 Integration By Partial Fractions 4 Integration By Parts 1 Integration By Parts 2 Integration By Parts 4 Differential Equations Particular Solution Integration By Parts 3 Applications Of Differential Equations

Inverse Trigonometric Functions By reversing the process we get : We can extend this to the following standard integral : LEARN QUESTION : If

By reversing the process we get : We can extend this to the following standard integral : LEARN QUESTION : If

So we only need to know standard integrals for sin -1 x and tan -1 x

Example Find :

Example Find :

Example Find : We need to have to use the standard integral.

Example Find : Again we need to have to use the standard integral.

Integration Using Partial Fractions (1) Example Find : Factorise if possible

Multiply both sides by (2x-3) (x+2)

Example Find : Multiply both sides by (x-1) (x+1) (2x+1)

Integration Using Partial Fractions (2) Example Find : This is where the denominator has a repeated linear factor Multiply both sides by (2x+1)(x-1) 2

Integration Using Partial Fractions (3) Example Find : This is where the denominator has an irreducible quadratic Multiply both sides by (x-2)(x 2 +1)

In some cases the value of B or C is zero and this makes the question easier.

Example Find : Multiply both sides by (x+2)(x 2 +9)

Example Find : Multiply both sides by (x-1)(x 2 +16)

Integration Using Partial Fractions (4) Example Find : This is where we have an improper algebraic fraction. -x 2 + x 4x + 3 x 3 + 2x 2 + 3x + 1 x 2 +3x+2 x x 3 + 3x 2 + 2x -x 2 - 3x

Multiply both sides by (x+1)(x+2)

Integration By Parts (1) We already know how to differentiate products. To integrate them we use a method called : integration by parts. LEARN This uses the formula :

We differentiate u to get We integrate to get v From the L.H.S, to get the R.H.S. : Generally choose u so that is simpler than u.

Example If we let u = x, Find : If we let u = sin x, The one which makes easier is u = x

Example Find :

Sometimes the choice is made for us as there are some functions which we can differentiate but do not know how to integrate.

Example Find :

Integration By Parts (2) Sometimes we need to repeat the process of integration by parts as although the function is simplified, it is still a product. Example Find :

Integrate by parts again

Example Find :

Integration By Parts (3) For functions like ln x or sin -1 x where we can differentiate them but not integrate it. Write the function as : This is called integration by parts using a “dummy” variable.

Example Find :

Example Find : Because of the square root the derivative of the denominator cannot be written as a multiple of the numerator. This is not a standard derivative of a natural logarithmic function

The derivative of the term inside the square root sign : -2x is a multiple of the numerator : x To integrate we will use a substitution

Integration By Parts (4) This is where the integral does not become simpler but eventually returns to its original form. Example Find :

We have returned to the original integral To solve this type of questions we take all the integrals to one side and add a constant to the other side.

Example Find

Differential Equations A differential equation is an equation connecting x, y and differential coefficients. The order of the equation is determined by the largest power of the differential coefficients. 1 st Order 2 nd Order There are various types of differential equations : variable separable exact homogeneous linear We only deal with a small number of these in Adv. Higher

Example Solve : This is known as the general solution of the differential equation.

First Order Variable Separable In this type we can write the equation with everything containing x on one side and everything containing y on the other. Example Find the general solution of

Example Find the general solution of

Example Find the general solution of

Example Find the general solution of

Example Find the general solution of

Particular Solutions This is where we are given enough information to find the constant. Example Find the particular solution of given that In Higher it was usually a point on the curve. passes through the point (1, 10)

Particular solution

Example Find the particular solution of given that x = 0 when y = 5

Sometimes the question wants the answer explicitly for y in terms of x. y = ……

Example Find the particular solution of given that x = 0 when y = 0

Applications Of Differential Equations The method for solving differential equations was developed when the need arose. Daniel Bernoulli developed the Bernoulli Principle on fluid flow which is still used today in the development of aircrafts, ships, bridges and other structures where air or water pressure play a part.

The Tacoma Bridge “Galloping Gertie” Video Link 1 Video Link 2 However WHEN THE MATHS IS NOT CORRECT BAD THINGS HAPPEN

Newton’s Law Of Cooling If we take a cup of water at room temperature, the rate at which it cools is proportional to the difference between the two temperatures.

Example If T is the temperature difference between two objects at time t minutes and at t = 0, express T in terms of T 0

Example The rate at which a radioactive material decays at any instant is proportional to the mass remaining at that instant. Given that there are x grams present after t days the differential equation in x is modelled by The half life of the radioactive material is 25 days. Find the time taken for 100g to decay to 20g.

(Half life is 25 days)

Example A container is shaped so that when the depth of the water is x cm, the volume of water in the container is.Water is poured into the container so that when the depth of water is x cm, the rate of increase is b) Solve the differential equation to obtain t in terms of x given that initially the container is empty.