A2 Differentiation and Integration Methods.

Differentiation

Function / Differential / Example / Notes
/ / / If you differentiate a constant you get zero
/ / / You can differentiate polynomials term by term.
/ / / Multiply by the power of the bracket.
Multiply by the differential of the function in the bracket.
Drop the power of the bracket by 1.
/ / / This is the general formula that works for polynomials and later functions. (E.g. see sinnf(x))
/ / / Function differentiated divided by function.
/ / A special function whose gradient at any point is equal to the value of the function at that point.
/ / / e to the function of x multiplied by the differential of the function.
/ / When a=e, lne=1 so result reduces to that above for ex.
/ / / Differential of function multiplied by cos of function.
/ / / Differential of function multiplied by differential of sin of function.
/ / / Differential of function multiplied by differential of tan of function.
/ / / Multiply by the power.
Multiply by the differential of the function
Multiply by the differential if sin.
Drop the power by one.
/ / / Product rule
/ / / Quotient rule.

Integration

Function / Integral / Example / Notes
/ / / Raise the power by 1 and divide by the new power.
Does not work when n=-1, this give the special case lnx.
/ / / You can integrate polynomials term by term.
/ / / Realize that the differential of the bracket is outside the bracket.
/ / / This is the general formula.
Realize that the differential of the bracket is outside the bracket.
/ / Consider using the basic rule for integration. This would imply that the area is always infinite beneath a 1/x graph, which is clearly ridiculous. Hence the special case.
/ / / Spot, function differentiated divided by function.
/ / Since the differential of ex is ex, then the integral of ex is ex (+c).
/ / / e to the ax divided by a..
/ / If a=e this reduces to the ex result above.
/ / ln can only take positive values.
/ / / Integrating sinnx and cosnx is relatively easy compared to integrating powers of sin and cos.
/ / / Recognize that you have a function multiplied by its differential.
/ / Notice similarity between sin and cos forms.
/ / Integration by parts.

Products of Trig Functions

Integral / Method / Example
1. òsinn x cosm x dx, n is odd / Factorize one sine out and convert the remaining sines to cosines using sin2 x =1- cos2 x. /
2. òsinn x cosm x dx, m is odd / Factorize one cosine out and convert the remaining cosines to sines
using cos2 x =1- sin2 x.
3. òsinn x cosm x dx n and m are both odd / Use either 1. or 2.
4, òsinn x cosm x dx n and m are both even. / Use double angle formula for sine and/or half angle
formulas to reduce the integral into a form that can be integrated.
1. ò tann x secm x dx, n is odd. / Factorize one tangent and one secant out and convert the remaining tangents to secants using tan2 x= sec2 x -1.
2. ò tann x secm x dx, m is even. / Factorize two secants out and convert the remaining secants to tangents using sec2 x =1+ tan2 x . /
3. ò tann x secm x dx, n is odd and m is even. / Use either 1. or 2.
4. ò tann x secm x dx, n is even and m is odd. / Each integral will be dealt with differently

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