Calc Formulas

Formulas from Calculus Derivatives [ex ]

= ex

[bx ]

= bx ln b

= 1

d dx d dx d dx

[ln x]

=

h i

= − x12

d dx

[logb x]

=

d dx d dx

h

= − x23

d dx d dx

[sinh x]

= cosh x

[cosh x]

= sinh x

d dx

h

d dx d dx d dx

[tanh x]

= sech 2 x

[arcsinh x]

=

d dx d dx d dx

[xn ]

= nxn−1

[c]

= 0

[x]

d dx

1 x

1 x2

i

√ [ x] √1 x

i

=

1 √ 2 x

= − 2x1√x

[arctanh x] =

[sin x]

= cos x

[cos x]

= − sin x

1 x

d dx d dx d dx

[tan x]

= sec2 x

1 x ln b

d dx

[sec x]

= tan x sec x

d dx d dx

[arcsin x]

=

√ 1 1+x2 1 1−x2

[f (x)g(x)] = f 0 (x)g(x) + f (x)g 0 (x)

Product Rule:

d dx

Quotient Rule:

d f (x) g(x)f 0 (x) − f (x)g 0 (x) = dx g(x) g(x)2

Chain Rule:

d [f (g(x))] = f 0 (g(x))g 0 (x) dx



Special Cases d [f (x)n ] dx d 1 dx g(x) 

=

d [ln |f (x)|] = dx d h f (x) i e dx



= nf (x)n−1 f 0 (x)



[arctan x] =

−g 0 (x) g(x)2 f 0 (x) f (x)

= f 0 (x)ef (x)

or

dy dy du = dx du dx

√ 1 1−x2 1 1+x2

Integrals Z

n

x dx

=

1 dx Z x c dx Z

= =

=

sinh x dx

= cosh x + C

1 2 2x 1 3 3x

Z

+C

n

x ln x dx

=

Z

xn+1 n+1

= − cos x + C

cos x dx

= sin x + C

tan x dx

= ln | sec x| + C

sec x dx

= ln | tan x + sec x| + C

sin2 x dx

=

1 2 (x

− sin x cos x) + C

cos2 x dx

=

1 2 (x

+ sin x cos x) + C

Z

ln x −

Z Z Z

xn+1 (n+1)2

+C

Z

tan2 x dx = tan x − x + C

Z

sec2 x dx

f (g(x))g (x) dx =

f (u) du = F (u) + C = F (g(x)) + C Z g(b)

0

f (u) du

f (g(x))g (x) dx = g(a)

a

Z

f 0 (x) dx = ln |f (x)| + C f (x)

Z

ef (x) f 0 (x) dx = ef (x) + C

By parts

u dv = uv −

Z b

u dv = a

Z

or Z b

uv]ba

Z

v du −

Z b

v du a

0

f (x)g (x)dx = f (x)g(x) − 0

= tan x + C

Z

0

Z b

sin x dx Z

1 √ dx x Z 1 dx = arctan x + C 1 + x2 Z 1 √ dx = arcsin x + C 1 − x2 Z ln x dx = x ln x − x + C

Z

1 x ln b b

cosh x dx = sinh x + C

+C

= −

Z

f (x)g (x)dx = a

bx dx

Z

1 +C x √ = 23 x x + C √ = 2 x+C

1 dx 2 Z x √ x dx Z

Special cases

= ex + C

Z

= cx + C

x2 dx

Substitution

ex dx

+C

Z

x dx

Z

Z

= ln |x| + C

Z Z

1 n+1 n+1 x

f (x)g(x)]ba

Z

−

g(x)f 0 (x) dx Z b a

g(x)f 0 (x) dx