Integration

Question 1

Sum rule

Answer 1

If f(x) = g(x) + n(x), 
Then ∫ f(x) dx = ∫ g(x) dx + ∫ n(x) dx

• Integrate term by term

Question 2

Constants rule

Answer 2

∫ ax^n dx = ax^(n+1) / (n+1)

• Leave constants out of integration, then multiply in at the end

Question 3

Calculating the constant of integration

Answer 3

1. Integrate to obtain the original function
2. Substitute in given values from question
3. Solve equation to find ‘c’ value
4. Rewrite integrated function with ‘c’ value (and the original x/y variables)

• If beginning from the second derivative, integrate twice to obtain the original function. This will produce two constants of integration.

Question 4

Integrating exponential functions

Answer 4

The answer will be the original function divided by the differentiated value of the power, + c ( x 1/x’)

• Leave constants out of integration, then multiply in at the end

Question 5

Integrating log functions

Answer 5

1. Break down main fraction into smaller fractions
2. Apply sum rule
3. Integrate each term separately
4. Rewrite integrated function, + c

• Terms on the bottom half of the fraction can be moved to the top by inverting the sign of the power

Question 6

Integrating trigonometric functions

Answer 6

The answer will be the integrated trig function divided by the differentiated value of the ‘x’ value, + c ( x 1/x’)

• Leave constants out of integration, then multiply in at the end

Question 7

Integrating trigonometric products

Answer 7

1. Rewrite function to be integrated as sums (using trig product rules)
2. Apply sum rule
3. Integrate each term separately
4. Simplify and rewrite integrated function, + c

• Leave constants out of integration, then multiply in at the end

Question 8

Guess/check method for integrating products

Answer 8

If question is f [g(x)]^n
- Try differentiating g(x)^(n+1)

If question is f (e)^g(x)
- Try differentiating (e)^g(x)

If question is f [trig(g(x))]
- Try differentiating ‘differentiation result’ of trig (g(x))

Then from the result of the guess, determine the factor needed to reach the same guess value tried. Then, add this factor to the guess value tried and rewrite function, + c

Question 9

Guess/check method for integrating quotients

Answer 9

If question is f’(x)/f(x)
- Try differentiating f(x)

Then from the result of the guess, determine the factor needed to reach the same guess value tried. Then, add this factor to the guess value tried and rewrite function, + c

Question 10

Integrating rational functions in the form (ax+b) / (cx + d)

Answer 10

Divide the denominator into the numerator, then integrate

Question 11

Integrating by substitution

Answer 11

1. Write the expression to be integrated in terms of u instead of x
2. Adjust for integrating with respect to u instead of x.
   - u = term
   - du/dx = differentiated term
   - du = differentiated term (dx)
   - dx = (du)/differentiated term
   - In the integral, after making the substitution with u, replace dx with its equivalent value
   eg. (du)/differentiated term
3. After completing the integration ‘substitute back’, so the final answer is in terms of x, not u.

Question 12

Properties of definite integrals - Sign

Answer 12

1. b∫a f(x) dx = -a∫b f(x) dx
   - Width of each integral is now (b-a)/n, hence the base of each rectangle is negative. If the order of the limits of integration are reversed, then the sign of the integral changes.

Question 13

Properties of definite integrals - Width = 0

Answer 13

2. a∫a f(x) dx = 0

- Width of each integral is 0.

Question 14

Properties of definite integrals - Additive

Answer 14

3. b∫a f(x) dx + c∫b f(x) dx = c∫a f(x) dx

- Area is additive

Question 15

Properties of even functions

Answer 15

• A function f(x) is even if f(-x) = f(x) for all values of x
• Function has the y-axis as an axis of symmetry

Question 16

Properties of odd functions

Answer 16

• A function f(x) is odd if f(-x) = -f(x) for all values of x
• Function has point symmetry (half-turn rotational symmetry) about the origin

Question 17

Properties of periodic functions

Answer 17

• A function f(x) is periodic if f(x) = f(x+a) for all values of x and some fixed, non-zero value ‘a’.
• Function repeats itself at regular intervals
• Smallest positive value of the fixed number ‘a’ is the period of the function.

Question 18

Integration properties of even functions

Answer 18

If a∫-a f(x) dx = 2 a∫0 f(x) dx,

Then function is even

Question 19

Integration properties of odd functions

Answer 19

If a∫-a f(x) dx = 0,

Then function is odd

Question 20

Areas with the x-axis as a boundary

Answer 20

b∫a f(x) dx gives the area between f(x) and the x-axis, bounded on the left by a vertical line x = a, and bounded on the right by the vertical line x = b (b is larger value).

Question 21

Signed areas (areas below the x-axis)

Answer 21

b∫a f(x) dx gives the area between f(x) and the x-axis, bounded on the left by a vertical line x = a, and bounded on the right by the vertical line x = b (b is larger value).

• Result will be negative, but give positive final answer as area cannot be negative

Question 22

Areas above and below the x-axis

Answer 22

Calculate the areas above and below the x-axis separately, then add them

• Results for areas below the x-axis will be negative, but give positive final answer as area cannot be negative

Question 23

Areas between two curves

Answer 23

b∫a [f(x) - g(x)] dx gives the area enclosed by two curves f(x) and g(x), and the lines x = a and x = b, where the function f(x) is above g(x).

Question 24

Areas with the y-axis as a boundary

Answer 24

1. Rewrite the curve as a function of y (x = ___)

2. Integrate with respect to y

Question 25

1. Trapezium rule

Answer 25

The approximated area (T) is given by T = h/2 [y0 + 2y1 + 2y2 + ... + 2yn-1 + yn] Where, h is the interval length (different between successive x values) n is number of trapezia

Question 26

Using trapezium/simpsons rules if not given x/y values

Answer 26

Evaluate function values of x between the limits of integration in steps of 0.1

Question 27

2. Simpsons rule

Answer 27

The approximated area (S) is given by S = h/3 [y0 + 4y1 + 2y2 + 4y3 + 2y4 + 4y5 + ... + 2yn-2 + 4yn-1 + yn] Where, h is the interval length (different between successive x values) n+1 is the number of y-values (number of y-value must be odd, therefore the number of different intervals must be even)

Question 28

Differential equation for when the rate of change is proportional to x

Answer 28

dy/dx = kx

Question 29

Differential equation for when the rate of change is inversely proportional to x

Answer 29

dy/dx = k/x

Question 30

Differential equation for when the rate of change is proportional to the amount itself

Answer 30

dy/dx = ky

Question 31

Checking solutions of differential equations

Answer 31

1. Calculate the derivative(s) of the function by integrating both sides 2. Substitute the function and its derivative(s) into the differential equation to check that it is true

Question 32

Solving differential equations in the form dy/dx = f(x)

Answer 32

1. Use integration on both sides 2. If the equation is a simple second-order differential equation, integrate both sides twice. This gives a general solution that has two constants.

Question 33

Solving first-order differential equations in the form dy/dx = f(x) x g(x)

Answer 33

1. Separate the variables by writing the equation with the x-terms on one side and the y-terms on the other 2. Use integration on both sides

Question 34

The constant in differential equations

Answer 34

1. Only one constant turns up in the solution of a first-order differential equation 2. When integrating expressions that have a log function as their anti-derivative (eg. 1/x), the constant of integration can be a multiplicative one (k) inside the logarithm function rather than additive (+c)

Question 35

Obtaining particular solutions from differential equations when given boundary conditions

Answer 35

1. Solve the differential equation 2. Substitute x and y values from equation into the equation and solve to find 'c' 3. Rewrite equation with 'c' value (and the original x/y variables)

Question 36

Applications of differential equations

Answer 36

1. Solve the equation, introducing a constant as part of the solution 2. Use the information given to evaluate the constant and substitute back to obtain a particular solution 3. Use the particular solution to answer some question

Question 37

Application differential equation for when a quantity is changing at a steady rate

Answer 37

dy/dt = k, y = kt + c

Question 38

Application differential equation for when the rate of change of a quantity is some function of time

Answer 38

dy/dt = f(t), y = ∫ f(t) dt

Question 39

Application differential equation for when the rate of change of a quantity is proportional to the quantity itself

Answer 39

dy/dt = ky, y = Ae^kt

Question 40

Application for when the differential equation has separate variables

Answer 40

Separate the variables and then integrate

Question 41

Newtons law of cooling

Answer 41

dT/dt = -k(T - To) Where, T is the temperature of the hot body To is the temperature of its surroundings

Question 42

Application of Newtons law of cooling

Answer 42

T = To + Ke^-kt - Two conditions (T and t) are required to solve for the two constants involved (k and K)

Question 43

Integral of tan(x)

Answer 43

-ln |cos(x)|

Question 44

Integral of 1/tan(x)

Answer 44

ln |sin(x)|

Question 45

Equivalent of tan(x)

Answer 45

sin(x)/cos(x)