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2 ways: integral ∫(x-a)/(x+a) dx
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2nd/3ways: trig integral ∫sec(x) dx
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The University of Maine: rational function integral compute
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2 ways: derivative x √√x
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1st way: count values nPn, nP(n-1), nP(n-2)
Alternate way: 2nd picture Show 1+3+5+...+(2n-1)=n^2
2 ways: trig show tan(2x)=2tan(x)/1-tan^2(x))
5th/5ways: triangular num: Show t(n-1) + tn = n^2 where tn = nth triangular number
log sq root integral ∫ln(√ x^3)/√ √ x dx
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examples 2x2 matrix withOUT an inverse
Find the derivative of f(x)=(1+2x^2)(x-x^2) in 2 ways
3rd/3ways: derivative (x+1)^(1/2) (x+2)^(1/3) (x+3)^(1/4) (x+4)^(1/5) (x+5)^(1/6)
exp trig derivative x^x sin(x)
UC Davis P#6: 2nd way differentiate y= (√x)^(√x) e^(x^2)
exp integral ∫(e^x +1)^3 dx
alternate way: count sum 1+2+3+ ... +n
exp derivative (log f(ax+b) (g(cx+d)))^h(mx+n)
2 ways: Find the derivative y=x^n logb(x)
2 ways of "counting" (a+b)^2 = a^2+2ab+b^2
2 ways: integral ∫x/(x+a) dx