Integration 2a

BAT Find integrals using rule for log functions Integration I KUS objectives BAT Find integrals using rule for log functions BAT Solve problems where partial factions used with this rule; Starter: Differentiate 2𝑥−3 7 Integrate 2𝑥−3 7 Integrate 𝑒 4𝑥+7

Therefore you can use either x or –x in the Integral   Notes As we are integrating to find the Area, you can see for any 2 points, the area will be the same for either graph… Therefore you can use either x or –x in the Integral However, you cannot find ln of a negative, just use the positive value instead!     This is saying when we Integrate either of the following, we get the same result:      

WB 5 Find the following integral 1 3𝑥 + 2 𝑑𝑥   1) Integrate the function using what you know from differentiation 2) Divide by the coefficient of x   3) Simplify if possible and add C We can extend the rule 𝑓′(𝑎𝑥+𝑏) 𝑑𝑥= 1 𝑎 𝑎𝑥+𝑏 +𝐶 To 𝑓 ′ 𝑥 𝑓 𝑥 𝑑𝑥= ln 𝑓(𝑥) +𝐶

WB 6 Find the following integrals: a) 1 4𝑥−5 𝑑𝑥 b) 3 3𝑥+11 𝑑𝑥 c) −2 7−6𝑥 𝑑𝑥 1 4𝑥−5 𝑑𝑥 = 1 4 ln 4𝑥−5 +𝐶 3 3𝑥+11 𝑑𝑥 = 1 3 3 ln 3𝑥+11 +C = ln 3𝑥+11 +𝐶 −2 7−6𝑥 𝑑𝑥 = 1 −6 −2 ln 7−6𝑥 +C = 1 3 ln 7−6𝑥 +𝐶

WB 7 𝑓 ′ 𝑥 = 5 5𝑥−1 , Given that the curve 𝑓(𝑥) passes through point (8, ln 16 ) find 𝑓(𝑥) Using point (8, ln 3 ) ln 16 = ln 8 +𝐶 𝐶 = ln 16 − ln 8 = ln 2 So (8, ln 3 ) 𝑓 𝑥 = ln 5𝑥−1 + ln 2

WB 8 Show that 2 7 2 4𝑥−3 𝑑𝑥 = ln 5 2 7 2 4𝑥−3 𝑑𝑥 = 1 4 2 ln 4𝑥−3 7 2

Skills 212 homework 212

WB 9 Find the following integral 𝑥−5 𝑥+1 𝑥−2 𝑑𝑥         Let x = 2         Let x = -1                      

WB 10a Find the following integral 9 𝑥 2 −3𝑥+2 9 𝑥 2 −4 𝑑𝑥 by rearranging to the form 𝑁+ 𝐴 3𝑥+2 + 𝐵 3𝑥−2                                              

Find the following integral: WB 10b   Find the following integral:                 =𝑥+ 1 3 ln 3𝑥−2 3𝑥+2 2 +𝐶

WB 11a Find the following integral 3 𝑥 2 +4𝑥−11 (𝑥−2) (𝑥+1) 2 𝑑𝑥 by rearranging to partial fractions 3 𝑥 2 +4𝑥−11 (𝑥−2) (𝑥+1) 2 = 𝐴 𝑥−2 + 𝐵 𝑥+1 + 𝐶 (𝑥+1) 2 =𝐴( 𝑥+1 2 +𝐵 𝑥−2 𝑥+1 +𝐶(𝑥−2) 𝑥=2 gives 9A= 12+8-11 solves to 𝐴=1 𝑥=−1 gives -3C=3-4-11 solves to C=4 𝑥=0 gives A-2B-2C=-11 solves to B=2 3 𝑥 2 +4𝑥−11 (𝑥−2) (𝑥+1) 2 = 1 𝑥−2 + 2 𝑥+1 + 4 (𝑥+1) 2

= 1 𝑥−2 + 2 𝑥+1 + 4 (𝑥+1) 2 𝑑𝑥 = ln 𝑥−2 +2 ln 𝑥+1 −4 𝑥+1 −1 +C WB 11b Find the following integral 3 𝑥 2 +4𝑥−11 (𝑥−2) (𝑥+1) 2 𝑑𝑥 by rearranging to partial fractions = 1 𝑥−2 + 2 𝑥+1 + 4 (𝑥+1) 2 𝑑𝑥 A knotty integral! ‘KNOT’ a logarithm term = ln 𝑥−2 +2 ln 𝑥+1 −4 𝑥+1 −1 +C

WB 12 Show that 5 6 3𝑥+2 (𝑥+3)(𝑥−4) 𝑑𝑥= ln 9 2 3𝑥+2 (𝑥+3)(𝑥−4) = …= 1 𝑥+3 + 2 𝑥−4 5 6 3𝑥+2 (𝑥+3)(𝑥−4) 𝑑𝑥 = 5 6 1 𝑥+3 + 2 𝑥−4 dx = ln 𝑥+3 + 2 ln 𝑥−4 6 5 = ln 9 +2 ln 2 − ln 8 + 2 ln 1 = ln 9 + ln 4 − ln 8 = ln 9×4 8 = ln 9 2

WB 13 Show that 𝑒 2 𝑒 4 5𝑥−4 𝑥 2 −𝑥 𝑑𝑥=8+ln 𝑒 4 −1 𝑒 2 −1 5𝑥−4 𝑥 2 −𝑥 = …= 4 𝑥 + 1 𝑥−1 𝑒 2 𝑒 4 5𝑥−4 𝑥 2 −𝑥 𝑑𝑥 = 𝑒 2 𝑒 4 4 𝑥 + 1 𝑥−1 𝑑𝑥 = 4 ln 𝑥 + ln 𝑥−1 𝑒 4 𝑒 2 = 4 ln 𝑒 4 + ln 𝑒 4 −1 − 4 ln 𝑒 2 + ln (𝑒 2 −1) = 16 + ln 𝑒 4 −1 −8 − ln (𝑒 2 −1) = 8+ln 𝑒 4 −1 𝑒 2 −1

= 1 𝑥−2 + 2 𝑥+1 + 4 (𝑥+1) 2 𝑑𝑥 = ln 𝑥−2 +2 ln 𝑥+1 −4 𝑥+1 −1 6 3 WB 14 Show that 3 6 3𝑥+2 (𝑥+3)(𝑥−4) 𝑑𝑥= ln 49 4 − 11 7 by rearranging to partial fractions From WB11 earlier = 1 𝑥−2 + 2 𝑥+1 + 4 (𝑥+1) 2 𝑑𝑥 = ln 𝑥−2 +2 ln 𝑥+1 −4 𝑥+1 −1 6 3 = ln 4 +2 ln 7 − 4 7 − ln 1 +2 ln 4 − 4 4 = ln 4 +2 ln 7 − 4 7 −0−2 ln 4 −1 = 2 ln 7 − ln 4 − 11 7 = ln 49 4 − 11 7 QED

Skills 213 homework 213

KUS objectives BAT Find integrals using rule for log functions BAT Solve problems where partial factions used with this rule; self-assess One thing learned is – One thing to improve is –

WB 15 Integrate the following:   WB 15 Integrate the following:   Start by trying y = ln|denominator|   Differentiate   This is double what we want so multiply the ‘guess’ by 1/2    

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