Core 3 - Integration

Question 1

What must you remember when you integrate to a ln?

Answer 1

It must be modulus, since a negative value is not defined on a ln graph, hence the absolute value must be taken.

Question 2

Integration by Substitution:

Integrate x(x+4)/(x-3)^2
use substitution u=x-3

Answer 2

1) du/dx = 1 => du=dx

2) express x+4 and X in terms of u
=> (u+3)(u+7)/u^2

3) expand brackets
=>(u^2 + 10u + 21)/u^2

4) separate brackets and cancel
=> 1 + 10/u + 21u^-2

5) Integrate each part
=> u + 10ln(u) - 21u^-1 +c

6) replace u with x term
=> (x-3) + 10ln(x-3) - 21(x-3)^-1 +c

Question 3

What is the rule for choosing u when integrating by parts?

Answer 3

U:LATE

Question 4

What is the formula for the integral based on integrating by parts?

Answer 4

i=uv- i(v(du/dx))

Question 5

When integrating between limits, what must you do in order not to get the wrong answer?

Answer 5

Change the limits by subbing the old limits in terms of x into the expression for u

e.g. if u=(3x^2 + 5), between 2 and 1,

The new limits are 17 and 8

Question 6

What can be taken out of an integral to make life easier?

Answer 6

any integer values. For example, when differentiating u in substitution, du/dx = 2, => du=2(dx).

Take the 2 outside the integral

Question 7

What can be combined when integrating by substitution to make life easier?

Answer 7

Combine any x terms in du/dx and in the question

y=x(x^2+3)

du/dx = 2x. Use this x to tidy up the first x term in the equation

0.5 f (x^2+3)

Question 8

What technique can help to tidy up an integral?

Answer 8

Take all integers outside brackets, and combine terms. This may include adding powers and expanding brackets

(u-1/3) * u^0.5

Expand brackets:
u*u^0.5 = (u^3/2-u^0.5)/3)

Take 1/3 outside the brackets

Question 9

When converting between dx and du, what can we look to combine?

Answer 9

Any part of the integral which cancels with part of the expression for dx

u=1+2tanx
du/dx = 2sec^2(x)
du=2sec^2(x) dx

the sec^2(x) cancels with the 1/cos^2(x) so this can simply be written as du/2.

Question 10

How do you find the limits without having to change the expression back to being in terms of x?

Answer 10

Sub the x values into the expression for u. This will give 2 limits in terms of u which you can keep in your integral?

Question 11

What is the equation for a volume of revolution around the x axis?

Answer 11

(i)=(pi)[y^2] between b and a

where y is the radius.

Question 12

What is the equation for volume of revolution around the y axis?

Answer 12

(i)=(pi)[x^2] between b and a

where x is the radius. This means you may have to rearrange.

Question 13

If you are rearranging an equation for x, in which order must you do it to eventually find x^2?

Answer 13

Rearrange for x first, then square the function.

Question 14

What do you need to remember when finding the volume of revolution of a solid?

Answer 14

Square the equation that gives the radius.

Question 15

When finding the indefinite integral, what must you remember at the end? Why?

Answer 15

+c. This is he constant of integration. Because we don’t have limits.

Question 16

When integrating by parts twice, what should you use to ensure you get the signs right?

Answer 16

Brackets around the second integral when substituting into the first.

Question 17

How can you use reverse chain rule to integrate a function like y=tan(x)?

Answer 17

Rewrite as y=sin(x)/cos(x), then rewrite it as -(int)(-sin(x))/cos(x), hence bottom differentiates to top, then write as ln(sec^2(x))

Question 18

What type of integral is reverse chain rule useful for?

Answer 18

Polynomials e.g. (2x+3)^7

Question 19

What is the formula for finding solid of revolution about the y axis?

Answer 19

π∫_x^X▒〖x^2 dy〗

Question 20

What is the formula for finding solid of revolution about the x axis?

Answer 20

π∫_x^X▒〖y^2 dX〗