Integrating something with xsquared on bottom of a fraction
Split the fraction up, and do it that way. Usually lnx + 3x to the -2.
Fraction with ax+b on both sides.
Factorise the top by using the bottom in the fraction. Add or sub on the end so it’s the same as it was originally. Split fraction up & simplify & integrate.
f’(x) / f(x)
Differentiate the normal, and if it matches the top, write ln(x).
If it doesn’t, divide by whatever.
Find the distance travelled in the 5th second
Write the seconds out!
[x=5] - [x=4]
Area under curves if there is -ve & +ve
Add the positives & minus the -ves
f(x) = something with y & k, etc.
Means that the rate of growth/decay is proportional to the amount.
If it’s not a fraction, 1/y dy, and kdx.
1/y integrates to lny
Sin squared x integrated
/
Cos squared x integrated
x/2 - 1/4(sin2x) + c
If it’s 2x, then it’ll be -1/8sin(4x)
Cos is the same but +
What is the third root x
x to the 1/3
Find the area between 2 curves
Top - bottom. (x1 - x2) - (x3 - x4)
Area under curve and above x axis
Just do the integral of the equation, biggest x - smallest. Do the working, then put it all in the calc to do it 4 u
How & where to use substitution
Use if when before integrating where the denominator is tough. Put u as the denominator.
Then do du/dx.
Rearrange to see what dx is.
Back in the fraction, put whatever / u and sub whatever in for dx.
Simplify, integrate, solve
2 equations & need to find the area between them, but aren’t given the point where they touch
Go into graph on calc and put the 2 equations in. Click intersection
Then do the integral (top) - (bottom) dx, then put the x values in.
Fraction: sec squared x/ tanx
Can write as ln(tanx)
Finding area under curve where part is above x-axis, and part is below.
Integral of negative part + integral of positive part.
Meaning area of negative + area of +ve.
Make sure to use the absolute (+ve value) of the -ve part. The 1st part of this and the last part of the other should be the same.
There are 2 areas under a curve, show that they’re equal or whatever
Find the total area, and show that the other is half that amount. They use the same equation, so u don’t have to integrate 2x
When integrating a fraction with x squared +c on the bottom
Can’t split the fraction up, use u sub
When integrating a fraction with a power on the bottom
First thing - bring it up! Don’t integrate w it on the bottom. Or if you do, minus the power
When doing the f’(x) = kt thing, and the starting temp is 25
Do f’(x) = k(t-25)
Quadratic on top of a fraction, ax+b on the bottom
Factorise twice on the top
(ax+b) / (cx+d)
Factorise the top by using the denominator.
Get the top to match the original, +- something on the end.
The number b4 the brackets gets removed, while the number after is still divided by the denom
If something resembles (f’x) / f(x)
Do ln of the bottom number
Multiply before the ln sign, or divide using fraction to match the differential of the top
If it’s dh/dt, and it = something with h
It’s like if dy/dx = y.
Change it around like with the kx stuff
