MATH 221

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updated: 18 August 2014

Lecture 22


f dfdx dfdx dfdx dx f The integral with respect to x undoes the derivative with respect to x. 5x4dx=x5, sincedx5dx =5x4. ALSO 5x4dx= x5+2,since d(x5+2)dx =5x4. This is like 9=3 and 9=-3.

We usually write 5x4dx=x5+c, where c is a constant, so that we can write all possible answers to 5x4dx at once. x2dx = x33 +c,sinced(x33)dx =133x2=x2. x3dx = x44+c,since d(x44)dx =144x3=x3. x4dx = x55+c,since d(x55)dx =155x4=x4. x5dx = x66+c,since d(x66)dx =166x5=x5. x642dx = x643643+c,since d(x643643)dx =1643643x642=x642. x-2dx = x-1-1+c,since d(x-1-1)dx =1-1(-1)x-2 =x-2. x-3dx = x-2-2+c,since d(x-2-2)dx =1(-2)(-2)x-3 =x-3. x-642dx = x-641-641+c,since d(x-641-641)dx =1-641(-641)x-642 =x-642. dx = 1dx= x0dx= x11+c,since dxdx=1. x-1dx = 1xdx= lnx+c,since dlnxdx= 1x. Note: x-1dx is NOT x00+c; x00 DOESN'T MAKE SENSE.

xdx= x12dx= x3232+c= 23x32+c, since d23x32dx= 2332x12=x12.

x3dx=x13 dx=x4343+c =34x43+c, since d34x43dx= 34·43x43-1 =x13.

x362431dx= x362431+1 362431+1 +c= x793431 793431 +c= 431793x793431+c, since d431793x793431dx= 431793793431x793431-1 =x362431.

exdx = ex+c,since dexdx=ex. e10xdx = e10x10+c,since d(e10x10)dx =110e10x·10=e10x. e-31xdx = e-31x-31+c, since d(e-31x-31)dx= 1-31e-31x (-31)=e-31x.

exdx= (eln2)xdx= exln2dx= exln2ln2+c= 2xln2+c, since d2xln2dx= dexln2ln2dx= 1ln2exln2·ln2= exln2=2x.

38xdx= (eln38)xdx= exln38dx= exln38ln38+c, since dexln38ln38dx= 1ln38exln38·ln38= exln38=38x.

sinxdx = -cosx+c,since d(-cosx)dx= -(-sinx)=sinx. cosxdx = sinx+c,since dsinxdx=cosx. sec2xdx = tanx+c,since dtanxdx= sec2x. csc2xdx = -cotx+c,since d(-cotx)dx =-(-csc2x)= csc2x. tanxsecxdx = secx+c,since dsecxdx= tanxsecx. cotxcscxdx = -cscx+c,since d(-cscx)dx= -(-cotxcscx)= cotxcscx.

cosxsin2xdx = cosxsinx 1sinxdx = cotxcscxdx = -cscx+c.

11+cosxdx = 1(1+cosx) (1-cosx)(1-cosx)dx = 1-cosx1-cos2xdx = 1-cosxsin2xdx = ( 1sin2x- cosxsin2x ) dx = ( csc2x- cotxcscx ) dx = -cotx+cscx+c.

Notes and References

These are a typed copy of Lecture 22 from a series of handwritten lecture notes for the class MATH 221 given on October 30, 2000.

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