INTEGRALSfunction integral

f(x)dg(x)dx

f(x)g(x)−∫df(x)dx

g(x)dx

xn(n 6= −1) xn+1

n+11x `n|x| Note:- `n|x|+K = `n|x/x0|ex ex

sinx − cosx

cosx sinx

tanx `n| secx|cosec x −`n| cosec x+ cotx| or `n

∣∣tan x2

∣∣secx `n| secx+ tanx| = `n

∣∣tan(π4 + x

2

)∣∣cotx `n| sinx|

1a2 + x2

1a

tan−1 x

a

1a2 − x2

12a`na+ x

a− xor

1a

tanh−1 x

a(|x| < a)

1x2 − a2

12a`nx− ax+ a

or − 1a

coth−1 x

a(|x| > a)

1√a2 − x2

sin−1 x

a(a > |x|)

1√a2 + x2

sinh−1 x

aor `n

(x+√x2 + a2

)1√

x2 − a2cosh−1 x

aor `n|x+

√x2 − a2| (|x| > a)

sinhx coshx

coshx sinhx

tanhx `n coshx

cosech x −`n |cosechx+ cothx| or `n∣∣tanh x

2

∣∣sech x 2 tan−1 ex

cothx `n| sinhx|

Double integral ∫ ∫f(x, y)dxdy =

∫ ∫g(r, s)Jdrds

where

J =∂(x, y)∂(r, s)

=

∣∣∣∣∣ ∂x∂r

∂x∂s

∂y∂r

∂y∂s

∣∣∣∣∣