PARUL POLYTECHNIC

INSTITUTE (IST SHIFT)

SHORT QUESTION NOTES OF UNIT – III DIFFERENTIATION AND ITS

APPLICATION FOR 2ND SEMESTER STUDENTS

PREPARED BY:

APPLIED SCIENCE & HUMANITIES DEPARTMENT

1. 0)( cdxd

, c is constant 18.x

xdxd

21)( 11

c

ax

xadx 1

22sin

2. 1)( nn nxxdxd

19. 2

11xxdx

d

= 1

1cos cax

3. xxdxd cos)(sin 20.

22

22 )(ax

xaxdxd

12.

22 axxdx

4. xxdxd sin)(cos 21.

22

22 )(xa

xxadxd

= cax

a1sec1

5. xxdxd 2sec)(tan 22. )][log( 22 axx

dxd

= 11cos1 c

axec

a

6. xxxdxd tansec)(sec =

22

1ax

13. ca

adxae

xx

log

7. xecxecxdxd cotcos)(cos 1.

)1(;1

1

ncnxdxx

nn 14. cee xx

8. xecxdxd 2cos)(cot 2. cxdx1 15.

22 axdx

9. aaadxd

exx log)( 3. cxlox

x1

= caxx 22log

10. aaadxd

exx log)( 4. cxxdx sincos 16. 22 ax

dx

11.x

xdxd

e

1)(log 5. cxxdx cossin = c

axax

a

log

21

12.2

1

11)(sin

xx

dxd

6. cxxdx tansec2 17. 22 xa

dx

13.2

1

11)(cosx

xdxd

7. cxxdxec cotcos 2 = c

axax

a

log

21

14. 21

11)(tanx

xdxd

8. cxxdxx sectansec

= cxaxa

a

log

21

15. 21

11)(cotx

xdxd

9. cecxxdxecx coscotcos =

16. 1

1)(sec2

1

xxx

dxd

10.

cax

aaxdx 1

22 tan1 = 1

1cot1 cax

a

PARUL POLYTECHNIC INSTITUTE (1ST SHIFT) Subject: Advanced Mathematics (Group – 1and 2)

Topic 2 & 3 – Differentiation & Integration (Formula)

17.1

1)(cos2

1

xxxec

dxd

Note: 2bxadx

= cxab

ab

1tan1

18. cxxdx seclogtan 28.If an example is of the form xbaxba sin

1,cos

1

,

19. cxxdx sinlogcot xcxba sincos

1

then take tx

2tan , 11

2t

dtdx

20. cxecxdx2

tanlogcos Also 22

2

12sin,

11cos

ttx

ttx

= cxecx cotcoslog 29. ),(;sin Nnmxdxxconnm

21.

cxxdx

24tanlogsec

1.Take tx sin or tx cos when m&n both are odd. 2.Take tx cos when m is odd & n is even. 3.Take tx sin when m is even & n is odd. 4.Use xxxx 2cos1cos2&2cos1sin2 22 when m & n both are even

= cxx tanseclog

22. cxfdxxfxf )(log)()('

If integral is of the form cbxax

dxcbxax

dx22 ,

Where ),(,0)(' baxxf Then take ])[( 22 xacbxax & proceed And ],[,0)( baxxf If integral is of the form

cbxaxBAxdx

cbxaxBAx

22,

Take ncbxaxdxdmBAx

)( 2

23.

cn

xfdxxfxfn

n

1)]([)()]([

1'

Where 0)(,0)(,1 ' xfxfn

'& ff are continuous functions nbaxmBAx )2(

24. dxxfxfxf n )(...)()( 21 comparing on both the side we get mbBna

Am ,2

= dxxfdxxfdxxfn

)(...)()( 21 Now,

dxcbxax

baxmdxcbxax

BAx22

2 +

cbxaxdxn 2

25. Rkdxxfkdxxkf ,)()( =

cbxax

dxncbxaxm 22log

26. If )()( xFdxxf then )(1)( baxFa

dxbaxf Where 0&),(: aRdcf

PARUL POLYTECHNIC INSTITUTE (1ST SHIFT) Subject: Advanced Mathematics (Group – 1 and 2)

Topic 2 & 3 – Differentiation & Integration (Formula)

1. cxfdx

xfxf )(log)()(1

9.

aa

dxxafdxxf00

)()(

2. c

nxfdxxfxf

nn

11

1)]([)()]([ , 1n 10. ,)(2)(

0

aa

a

dxxfdxxf if )(xf is even

i.e. )()( xfxf = 0 , if )(xf is odd i.e. )()( xfxf

3. cbaxF

adxbaxf )(1)(

cxfedxxfxfe xx )()]()([ 1 4.

)()()]([)( aFbFxFdxxf ba

b

a

11. dxxafdxxfdxxfaaa

a 00

2

)2()()(

,)(20a

dxxf if )()2( xfxaf

= 0 , if )()2( xfxaf

5. b

a

b

a

dxxfkdxxkf )()(

0)( a

a

dxxf , a

b

b

a

dxxfdxxf )()(

6. b

a

b

a

b

a

xgxfdxxgxf )()()]()([ 12. cxxdx seclogtan

7. b

a

b

a

b

a

dyyfdttfdxxf )()()( 13. cxxdx sinlogcot

8. b

c

c

a

b

a

xfdxxfdxxf )()()( 14. cxxxdx tanseclogsec

cx

24tanlog

15. cxcxecxecxdx2

tanlogcotcoslogcos 16.

dxVdx

dxdUVdxUUVdx

(1) The area bounded by the curve ),(xfy the X – axis and the lines x = a and x = b is given by

A = b

a

dxxf )(

(2) The area bounded by the curve ),(ygx the Y – axis and the lines y = c and y = d is given by

A = b

a

dyyg )(

(3) The volume of the solid generated by revolving about the X – axis, the area bounded by the curve

),(xfy and the lines x = a and x = b is given by V = b

a

dxxf 2)]([

(4) The volume of the solid generated by revolving about the Y – axis, the area enclosed by the curve

),(ygx and the line y = c and y = d is given by V = b

a

dyyg 2)]([

PARUL POLYTECHNIC INSTITUTE (1ST SHIFT) Subject: Advanced Mathematics (Group – 1 and 2)

Topic 2 & 3 – Differentiation & Integration (Formula)