integrals Flashcards
∫dx/ root a2- x2
sin inv (x/a )
∫dx/ax + b
1/a ln (ax+b)
∫a^(px+q)
∫ 1/p a(px+q)/lna
∫sin(ax+b)
-1/a cos(ax+b)
∫cos(ax+b)
i/a sin(ax+b)
∫tan (ax+b)
1/a ln (sec(ax+b))
∫cot(ax+b)
1/a ln (sin ax+b)
∫sec^2 ax+b
1/a tan ax+b
∫sec^2 (ax+b)
1/a tan(ax+b)
∫cosec^2 (ax+b)
-1/a cot(ax+b)
∫cosecx.cotx
cosecx
∫secx. tanx
secx
∫cosecx
ln(cosecx-cotx)
∫dx/ a2+ x2
1/a tan inv(x/a)
∫dx/ root x2+ a2
ln [x+ root x2+ a2]
∫dx/ x root x2- a2
1/a sec inv(x/a)
∫dx/ root x2 - a2
ln [x+ root x2- a2]
∫dx/ a2- x2
1/2a (ln mod a+x/a-x)
∫dx/ x2- a2
1/2a (ln mod x-a/ x+a)
∫root a2- x2 dx
(x/2) (root a2- x2) + (a2/2) (sin inv x/a)
∫root x2+ a2
(x/2) (root a2+ x2) + (a2/2) ln (x+ root x2+ a2)
∫root x2- a2
(x/2) (root a2- x2) - (a2/2) ln (x+ root x2- a2)
∫e^ax.sinbx dx
e^ax(acosbx+bsinbx)/a^2+b^2
∫e^ax.cosbx dx
e^ax(acosbx-bsinbx)/a^2+b^2
what should u substitute for ∫root (x-A/x-B)or ∫ (root x-A)(x-B)
put x= A sec2 theta - B tan2 theta
what should u substitute for ∫dx/ a2+ x2 or ∫root a2+x2
put x= a tan theta
what should u substitute for ∫dx/ a2- x2 or ∫root a2-x2
put x = a sin theta or a cos theta
what should u substitute for ∫dx/ x2- a2 or ∫root x2-a2
put x= a sec theta or a cosec theta
what should u substitute for ∫dx/ root (x-A)(x-B)
put x-A= t2 or x-B = t2
what should u substitute for∫root (a-x/a+x)
put x= a cos 2theta
what should u substitute for ∫root (x-A/B-x)or ∫ (root x-A)(B-x)
put x= Acos2 theta + B sin2 theta
∫u.v dx
u∫v - [∫ du/dx . ∫v dx]
∫[f(x)+ f’ (x)]
x f(x)
∫e^x [f(x)+ f’ (x)] dx
e^x. f(x)
∫ln (x)
x lnx - x
∫dx/ a+ b sin x or ∫dx/ a+ b cos x or ∫dx/ a sin x+ b sinx cosx + c cos x
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
∫sin^m (x) .cos ^n (x)
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
∫dx/ a+ b sin2 x or ∫dx/ a+ b cos2 x or ∫dx/ a sin2 x+ b sinx cosx + c cos2 x
divde N and D by cos2 x and put tanx= t
∫acosx+bsinx+c/pcosx+qsinx+r
express numerator= l(D) +mdD/dx +n
∫dx/x(x^n+1)
take x^n common and put 1+ x^-n = t
∫dx/x^2(x^n+1)^(n-1)/n
take n common 1+ x^-n = t^n
∫dx/x^n(x^n+1)^1/n
take x^n common and put 1+x^-n = t^n
∫dx/ (ax+b) rootpx+q or ∫dx/ (ax^2+bx+c) rootpx+q
put px+q = t^2
∫dx/ (ax+b) rootpx^2+qx+r
ax+b=1/t
∫dx/ (ax^2+bx+c) rootpx^2+qx+r
x=1/t
px^2+qx+r / (x-a)(x-b)(x-c)
A/x-a + B/x-b +C/x-c
px^2+qx+r / (x-a)^2(x-b)
A/x-a + B/(x-a)^2 +C/x-b
px^2+qx+r / (x-a)(x^2+bx+c)
A/x-a + Bx + C/x^2+bx+c
f(x)/ (x-a)(x^2+bx+c)
A/x-a + Bx+C/x^2+bx+c + Dx+ E/ (x^2 +bx+c)^2
∫dx/ (ax^2+bx+c) or ∫dx/ root(ax^2+bx+c) or ∫ root(ax^2+bx+c)
express (ax^2+bx+c) in the form of perfect sq and solve
∫px+q dx/ (ax^2+bx+c) or ∫px+q dx/ root(ax^2+bx+c)
express px+q = l (diff coeff of denominator) + m
∫x^2+1/x^4+ kx^2+1 or ∫x^2-1/x^4+ kx^2+1
divide n and d by x^2 and proceed
