integrals

Question 1

∫dx/ root a2- x2

Answer 1

sin inv (x/a )

Question 2

∫dx/ a2+ x2

Answer 2

1/a tan inv(x/a)

Question 2

∫dx/ root x2+ a2

Answer 2

ln [x+ root x2+ a2]

Question 2

∫dx/ x root x2- a2

Answer 2

1/a sec inv(x/a)

Question 3

∫dx/ root x2 - a2

Answer 3

ln [x+ root x2- a2]

Question 4

∫dx/ a2- x2

Answer 4

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

Question 5

∫dx x2- a2

Answer 5

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

Question 6

∫root a2- x2 dx

Answer 6

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

Question 6

∫root x2+ a2

Answer 6

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

Question 7

∫root x2- a2

Answer 7

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

Question 7

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

Answer 7

put x= A sec2 theta - B tan2 theta

Question 8

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

Answer 8

put x= a tan theta

Question 9

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

Answer 9

put x = a sin theta or a cos theta

Question 9

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

Answer 9

put x= a sec theta or a cosec theta

Question 9

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

Answer 9

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

Question 10

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

Answer 10

put x= a cos 2theta

Question 11

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

Answer 11

put x= Acos2 theta + B sin2 theta

Question 12

∫u.v dx

Answer 12

u∫v - [∫ du/dx . ∫v dx]

Question 12

∫[f(x)+ f’ (x)]

Answer 12

x f(x)

Question 12

∫e^x [f(x)+ f’ (x)] dx

Answer 12

e^x. f(x)

Question 13

∫ln (x)

Answer 13

x lnx - x

Question 13

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

Answer 13

convert sin and cos into half tan angles and put tan x/2= t

Question 14

∫sin^m (x) .cos ^n (x)

Answer 14

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 14

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

Answer 14

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

Question 15

∫