7.2B Natural Logarithms: Integration

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1 7.B Natral Logarithms: Intgration Sction 7.B Nots Pag n n Prviosly w lookd at th powr rl for intgration: d as long as n is not qal to -. Now w n will look at what happns whn n is -. W ar looking at this cas: d. Th following is how w intgrat this: Intgration of a Natral Logarithm Lt b a diffrntiabl fnction of. Thn:.) d C.) d C Sinc d d w can rwrit this as: d C W hav th absolt val symbols hr so that any val will fit th domain of th natral logarithm. EXAMPLE: Find th indfinit intgral: d. W will b sing sbstittion for most of th problms in this sction. Hr. Thn d d. Now w mak or sbstittion: d. Whn w intgrat this w gt C. Thn w can rplac th with and w gt: C as or answr. EXAMPLE: Find th indfinit intgral: d. d Using sbstittion w will gt. Thn d d. Solving for d w gt: d. Now w d mak or sbstittion:. This simplifis to: d. Whn w intgrat this w gt C. Thn w can rplac th with and w gt: C which is or answr.

2 Sction 7.B Nots Pag EXAMPLE: Find th indfinit intgral: 9 d. Using sbstittion w gt 9. Thn d d. Solving for d w gt: d d. Now w mak d or sbstittion:. This simplifis to: d. This can b rwrittn as: d Is a natral log ndd on this on? Th answr is no bcas th powr on th is not -. Whn w intgrat this w gt C. This simplifis to: C. Thn w can rplac th with 9 and w gt: 9 C which is or answr. EXAMPLE: Find th indfinit intgral: ( ) d. 4 Using sbstittion w will gt 4. Thn d 6 d. Solving for d and aftr factoring w d ( ) d gt: d. Now w mak or sbstittion: ( ). This simplifis to: d ( ). Whn w intgrat this w gt C. Thn w can rplac th with 4 and w gt: 4 C which is or answr. EXAMPLE: Find th indfinit intgral: 7 d. Th problm with this on is that if w s th nmrator for and thn tak th drivativ it won t cancl ot th dnominator. Thrfor w nd to s long division to brak this p into sparat prssions: 7 First w ask orslvs how many tims gos into. W ignor th - at th 4 momnt. W know gos into an amont of. W thn mltiply by to gt 4. W writ this on th nt lin and thn w sbtract. Thn w gt. W now ask how many tims dos go into. Th answr is 9. W thn mltiply by to gt. W sbtract and gt a rmaindr of 9.

3 Sction 7.B Nots Pag So now w know that 7 d = 9 d. Now w can intgrat ach part sparatly. For th last part, w will lt. Thn d d. Making th sbstittion w gt: 9 d. Intgrating w gt 9 C. Ptting this all togthr and intgrating th first two trms w will gt: 9 C as or answr. EXAMPLE: Find th indfinit intgral: d. Th problm with this on is that if w s th nmrator for and thn tak th drivativ it won t cancl ot th dnominator. Thrfor w nd to s long division to brak this p into sparat prssions. Rmmbr that all of th trms mst b ordrd in dscnding powrs and if thr is a missing trm yo mst pt in a zro plac kpr. 0 Rmmbr yo ar always sbtracting whn doing long division So or problm can b rwrittn as: d = d. Now w can intgrat ach d part sparatly. For th last part, w will lt d. So d d. Solving for d yo gt: d. d Making th sbstittion w gt:. Simplifying yo will gt: d. Intgrating w gt C. Ptting this all togthr and intgrating th first two trms w will gt: C. EXAMPLE: Find th indfinit intgral: Using sbstittion w will gt. Thn d d. Solving for d w gt: d d. Now w mak or sbstittion: d. d. This simplifis to: d. Whn w intgrat this w gt C. Thn w can rplac th with and w gt: C as or answr.

4 EXAMPLE: Find th indfinit intgral: d. Sction 7.B Nots Pag 4 Using sbstittion w will gt (). Thn d () () d by sing th chain rl. Simplifying yo will gt: d d. Solving for d w gt: d d. Now w mak or sbstittion: d. This simplifis to: d. W nd to mak anothr sbstittion to gt rid of th sqar root. W will tak or original dfinition for and solv for to gt:. Now w can mak anothr sbstittion: C d. This simplifis to: d. Whn w intgrat this w gt and w gt: C.. Thn w can rplac th with EXAMPLE: Find th indfinit intgral: tan d. sin W havn t don this on yt. First lt s s idntitis to writ this as: d. Now w can s cos sbstittion to intgrat this. W will lt cos. Thn d sin d. Solving for d yo will gt: d sin d d. Now w mak or sbstittion: sin. Simplifying will giv s: d sin. Intgrating this will giv: C. Thn w can rplac th with cos and w gt: cos C. W can do a similar procss for cot, sc, and csc to gt th following rslts: Intgrals of th Si Trigonomtric Fnctions sin d cos C cos d sin C tan d cos C cot d sin C sc d sc tan C csc d csc cot C Mor on nt pag

5 EXAMPLE: Find th indfinit intgral: sc d. Sction 7.B Nots Pag Using sbstittion w will gt. Thn d d. Solving for d w gt: d d. Now w mak or sbstittion: sc d. This simplifis to: sc d. Whn w intgrat this w gt sc tan C. Thn w can rplac th with and w gt: sc tan C. EXAMPLE: Find th indfinit intgral: csc cot d. W can intgrat ach thing sparatly by sing th intgration formlas: csc cot sin C sin W can s log proprtis to writ this as: C. This can also b simplifid by sing idntitis: csc cot sin sin sin C. This qals: C which simplifis to: C. W can s th cos cos cos sin sin sin cos cos cos idntity sin cos : C. Now w can factor th nmrator: C cos cos This simplifis to: cos C. EXAMPLE: Find th dfinit intgral: d. Using sbstittion w will lt. Thn d d. Solving for d w gt: d d. Now w mak or sbstittion: d. This simplifis to: d. Whn w intgrat this w gt. Thn w can rplac th with and w gt:. Now w plg in or top limit and or bottom limit:. This can b rwrittn as:. W also know that, so or problm bcoms:. Sinc 0 or answr is. EXAMPLE: Find th indfinit intgral: d. ( 4) ( 4) ( 4) Using sbstittion w will lt ( 4). Thn d d, so d d. Now w mak or 4 ( 4) sbstittion: d. This givs d c C ( 4) ( 4).

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST: .4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This