solution is, y = C.F + P.I = A cos ax +
2-12
(D2 - 2D + 2)y = 23x + Cos 2x
equation is (D2 - 2D + 2) y = e3x + Cos
ry equation is m2 - 2m + 2 = 0
Answers
A (t an- 1 x)=-i-
dx V ; 1 + x 2
d , -i , 1
— (sec x) = —
dx xV(x 2 - 1)
(v) — (sin h x) = cos h x
dx
dx
(tan h x) = sech 2 x
dx v v
u^_ vdu/dx — u dv/dx
— (ax + b) n =n(ax + b) n " 1 .a
dx
a x ) = a x log e a
dx
d n \ 1
— (log a x)= — .
dx x log a
d , ,
— (cos x) = — sin x
dx
d_
dx
_d_
dx
_d_
dx
(cot x) = — cosec 2 x
(cosec x) = — cosec x cot x.
(cos _1 x) =
-1
7(1 " x 2 )
- d -(cor 1 x)=-^ T
dx 1 + x^
d , _i . -1
— (cosec x) =
dx
d_
dx
_d_
dx
V(x 2 -1)'
(cos h x) = sin h x
(cot h x) = —cosec h 2 x.
(vi) D n (ax + b) m = m (m -1) (m - 2) (m - n + 1) (ax + b) m - n . a"
xiii
D n log(ax + b) = (- 1) (n - 1) ! a n /(ax + b) n
D n (e™) = m n e mx D n (a 1 ™) = m n (loga) n . a™
D n [?sf b r4]=(a 2+ *> 2 ) n/2
^ax
sin(bx + c + n tan b / a)
cos (bx + c + n tan - ^ b / a)
(vii) Leibnitz theorem: (uv) n
= U n + n ClU n -lVl+ n C 2 Un-2V2 + + n C r U n -rV r + + n C n V n .
3. Integration
,n + l
(i) fx" dx=^— (n
J n + 1
Je x dx = e x
(ii) Jsin x dx = — cos x
Jtan x dx = — log cos x
f f Tl X ^
Jsec x dx = log(sec x + tan x ) = log tan ^— + —
Jcosec x dx = log(cosec x — cot x ) = log tan ^
Jsec 2 x dx =
..... r dx 1 . _i x
( m ) J 2^ 2 = - tan 1 -
J a 2 + x z a a
f
f- dx = log e x
J X
Ja x dx = a x /log e a
Jcos x dx = sin x
Jcot x dx = log sin x
'a + x'
dx
1 , a + x
— lo 8
'a 2 — x 2 2a ° a - x
Jcosec 2 x dx = — cot x.
dx . _i x
==■ = sin —
a
p — , = sin h —
V(a 2 +x 2 ) a
f dx 1 . x — a
'x 2 - a 2 " 2a~ og 7T7
r dx
r dx
J V(x 2 -a 2 )
u-1 x
cos h — .
• -1 x