Core 3 - Integration Flashcards
What must you remember when you integrate to a ln?
It must be modulus, since a negative value is not defined on a ln graph, hence the absolute value must be taken.
Integration by Substitution:
Integrate x(x+4)/(x-3)^2 use substitution u=x-3
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
What is the rule for choosing u when integrating by parts?
U:LATE
What is the formula for the integral based on integrating by parts?
i=uv- i(v(du/dx))
When integrating between limits, what must you do in order not to get the wrong answer?
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
What can be taken out of an integral to make life easier?
any integer values. For example, when differentiating u in substitution, du/dx = 2, => du=2(dx).
Take the 2 outside the integral
What can be combined when integrating by substitution to make life easier?
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)
What technique can help to tidy up an integral?
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
When converting between dx and du, what can we look to combine?
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.
How do you find the limits without having to change the expression back to being in terms of x?
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?
What is the equation for a volume of revolution around the x axis?
(i)=(pi)[y^2] between b and a
where y is the radius.
What is the equation for volume of revolution around the y axis?
(i)=(pi)[x^2] between b and a
where x is the radius. This means you may have to rearrange.
If you are rearranging an equation for x, in which order must you do it to eventually find x^2?
Rearrange for x first, then square the function.
What do you need to remember when finding the volume of revolution of a solid?
Square the equation that gives the radius.
When finding the indefinite integral, what must you remember at the end? Why?
+c. This is he constant of integration. Because we don’t have limits.
When integrating by parts twice, what should you use to ensure you get the signs right?
Brackets around the second integral when substituting into the first.
How can you use reverse chain rule to integrate a function like y=tan(x)?
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))
What type of integral is reverse chain rule useful for?
Polynomials e.g. (2x+3)^7
What is the formula for finding solid of revolution about the y axis?
π∫_x^X▒〖x^2 dy〗
What is the formula for finding solid of revolution about the x axis?
π∫_x^X▒〖y^2 dX〗
