Integration Flashcards
What must we always remember for integration without limits
To add C (+C)
Basic rule for integration
Raise power by 1 and divide by new power
Chain rule for integration
When function with function we divide by the derivative of the function within. Or multiply by 1/derivative of function within.
§sinxdx=
-cosx +c
§cosxdx=
sinx +c
§e^xdx=
e^x +c
§e^3xdx=
1/3e^3x +c
What can we do to make integrals easier to evaluate
Take constants out the integral and deal with them after.
§1/3x+4dx
1/3 ln|3x+4| +c
Special rule for integration
§f’(x)/f(x)dx= ln(f(x) + c
What to look out for when doing integration involving a fraction
Special rule for integration, if top line is or can be derivative of top line
What can we do to help special rule for integration be the case
Manipulate the fraction then take multipliers outside the integral. Or add another fraction that makes the special rule work for one, and is still equal to same equation.
When integrating partial fractions for a Bx+C what should we do
Manipulate to use special rule for integration
Integration by substitution basic processes
Write out full integral.
For our u =f(x) we find du/dx and then write on one line, one side du=f’(x)dx we substitute in our initial value for u and also a du or 1/f’(x) du. Simplify our equation. Integrate. At the end re substitute our original variables.
What to remember for integration with definite integrals
We must change the definite integrals, we do this by subbing our definite integrals into u=x equation to work our u.
We use these in our square brackets to get our answer at the end.
Equation for integration by parts:
§uv’=uv-§u’v
In integration by parts u should ….
Get more simple and eventually differentiate to a constant.
Process of integration by parts.
Set up integral and evaluate which value should be u and which should be v’.
At side of page write down all values of u,v,u’ and v’. (We differentiate u and integrate v’).
Use our formula and follow it to conclusion. Remember to +c.
Integration by parts twice
If second integral is not simple then we may have to integrate again. Do same process, remember to bracket as it is being taken away.
What to do when integrating by parts with cyclic functions eg. (e^x and sin^x)
Set original integral = I.
Carry out integration by parts (usually twice), we may have to take a constant out of the integral. Substitute in I for integral. Move I to LHS so it’s a multiple I. Divide by multiple of I on LHS on both sides. Change c to C
