integrals

Question 1

∫dx/ root a2- x2

Answer 1

sin inv (x/a )

Question 2

∫dx/ax + b

Answer 2

1/a ln (ax+b)

Question 3

∫a^(px+q)

Answer 3

∫ 1/p a(px+q)/lna

Question 4

∫sin(ax+b)

Answer 4

-1/a cos(ax+b)

Question 5

∫cos(ax+b)

Answer 5

i/a sin(ax+b)

Question 6

∫tan (ax+b)

Answer 6

1/a ln (sec(ax+b))

Question 7

∫cot(ax+b)

Answer 7

1/a ln (sin ax+b)

Question 8

∫sec^2 ax+b

Answer 8

1/a tan ax+b

Question 9

∫sec^2 (ax+b)

Answer 9

1/a tan(ax+b)

Question 10

∫cosec^2 (ax+b)

Answer 10

-1/a cot(ax+b)

Question 11

∫cosecx.cotx

Answer 11

cosecx

Question 12

∫secx. tanx

Answer 12

secx

Question 13

∫cosecx

Answer 13

ln(cosecx-cotx)

Question 14

∫dx/ a2+ x2

Answer 14

1/a tan inv(x/a)

Question 15

∫dx/ root x2+ a2

Answer 15

ln [x+ root x2+ a2]

Question 16

∫dx/ x root x2- a2

Answer 16

1/a sec inv(x/a)

Question 17

∫dx/ root x2 - a2

Answer 17

ln [x+ root x2- a2]

Question 18

∫dx/ a2- x2

Answer 18

1/2a (ln mod a+x/a-x)

Question 19

∫dx/ x2- a2

Answer 19

1/2a (ln mod x-a/ x+a)

Question 20

∫root a2- x2 dx

Answer 20

(x/2) (root a2- x2) + (a2/2) (sin inv x/a)

Question 21

∫root x2+ a2

Answer 21

(x/2) (root a2+ x2) + (a2/2) ln (x+ root x2+ a2)

Question 22

∫root x2- a2

Answer 22

(x/2) (root a2- x2) - (a2/2) ln (x+ root x2- a2)

Question 23

∫e^ax.sinbx dx

Answer 23

e^ax(acosbx+bsinbx)/a^2+b^2

Question 24

∫e^ax.cosbx dx

Answer 24

e^ax(acosbx-bsinbx)/a^2+b^2

Question 25

what should u substitute for ∫root (x-A/x-B)or ∫ (root x-A)(x-B)

Answer 25

put x= A sec2 theta - B tan2 theta

Question 26

what should u substitute for ∫dx/ a2+ x2 or ∫root a2+x2

Answer 26

put x= a tan theta

Question 27

what should u substitute for ∫dx/ a2- x2 or ∫root a2-x2

Answer 27

put x = a sin theta or a cos theta

Question 28

what should u substitute for ∫dx/ x2- a2 or ∫root x2-a2

Answer 28

put x= a sec theta or a cosec theta

Question 29

what should u substitute for ∫dx/ root (x-A)(x-B)

Answer 29

put x-A= t2 or x-B = t2

Question 30

what should u substitute for∫root (a-x/a+x)

Answer 30

put x= a cos 2theta

Question 31

what should u substitute for ∫root (x-A/B-x)or ∫ (root x-A)(B-x)

Answer 31

put x= Acos2 theta + B sin2 theta

Question 32

∫u.v dx

Answer 32

u∫v - [∫ du/dx . ∫v dx]

Question 33

∫[f(x)+ f’ (x)]

Answer 33

x f(x)

Question 34

∫e^x [f(x)+ f’ (x)] dx

Answer 34

e^x. f(x)

Question 35

∫ln (x)

Answer 35

x lnx - x

Question 36

∫dx/ a+ b sin x or ∫dx/ a+ b cos x or ∫dx/ a sin x+ b sinx cosx + c cos x

Answer 36

convert sin and cos into half tan angles and put tan x/2= t…… sinx=2t/1+t^2
cosx= 1-t^2/1+t^2

Question 37

∫sin^m (x) .cos ^n (x)

Answer 37

case1; when one of them is odd then substitute for the term of even power
if both are odd then subtitute either of the term
if both are evn then convert everything into cos half angles
case2; m+n= negative even integer then substitute tanx= t

Question 38

∫dx/ a+ b sin2 x or ∫dx/ a+ b cos2 x or ∫dx/ a sin2 x+ b sinx cosx + c cos2 x

Answer 38

divde N and D by cos2 x and put tanx= t

Question 39

∫acosx+bsinx+c/pcosx+qsinx+r

Answer 39

express numerator= l(D) +mdD/dx +n

Question 40

∫dx/x(x^n+1)

Answer 40

take x^n common and put 1+ x^-n = t

Question 41

∫dx/x^2(x^n+1)^(n-1)/n

Answer 41

take n common 1+ x^-n = t^n

Question 42

∫dx/x^n(x^n+1)^1/n

Answer 42

take x^n common and put 1+x^-n = t^n

Question 43

∫dx/ (ax+b) rootpx+q or ∫dx/ (ax^2+bx+c) rootpx+q

Answer 43

put px+q = t^2

Question 44

∫dx/ (ax+b) rootpx^2+qx+r

Answer 44

ax+b=1/t

Question 45

∫dx/ (ax^2+bx+c) rootpx^2+qx+r

Answer 45

x=1/t

Question 46

px^2+qx+r / (x-a)(x-b)(x-c)

Answer 46

A/x-a + B/x-b +C/x-c

Question 47

px^2+qx+r / (x-a)^2(x-b)

Answer 47

A/x-a + B/(x-a)^2 +C/x-b

Question 48

px^2+qx+r / (x-a)(x^2+bx+c)

Answer 48

A/x-a + Bx + C/x^2+bx+c

Question 48

f(x)/ (x-a)(x^2+bx+c)

Answer 48

A/x-a + Bx+C/x^2+bx+c + Dx+ E/ (x^2 +bx+c)^2

Question 49

∫dx/ (ax^2+bx+c) or ∫dx/ root(ax^2+bx+c) or ∫ root(ax^2+bx+c)

Answer 49

express (ax^2+bx+c) in the form of perfect sq and solve

Question 50

∫px+q dx/ (ax^2+bx+c) or ∫px+q dx/ root(ax^2+bx+c)

Answer 50

express px+q = l (diff coeff of denominator) + m

Question 51

∫x^2+1/x^4+ kx^2+1 or ∫x^2-1/x^4+ kx^2+1

Answer 51

divide n and d by x^2 and proceed