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