crying

Sin²x =

½ (1-cosx)

Cos²x=

½ (1+cosx)

Sin(2x)=

2sinxcosx

Cos(2x)

Cos²x-sin²x , 1-sin²x , 2cos²x-1

If u see 1/(b²+a²x²)

Use x=b/a tanu, dx=cosudu

If u see √(b²-a²x²)

Use x=b/a sinu , dx=cosudu

If u see √(a²-x²)

Use x=asint dx=acost

If u see √(a²+x²)

Use x=atant , dx=asec²x

If u see √(x²-a²)

Use x=asecx , dx=asecxtanx

Int. Sec²x=

Tanx

Int. Csc²x=

-cotx

Int. Secxtanx=

Secx

Int. Csccotx=

-cscx

Int. Tanx=

Ln(secx)

Int. 1/√(1+x²)=

Sin^-1x

Int. -1/√(1-x²)=

Cos^-1x

Int. 1/1+x²=

Tan^-1c

Shell method

V= 2π int. x(f(x))

Area using polar coords

A= ½ int. r²

Arc length cartesian

L= int.√(1+f’(x)²)

Arc length parametrized functions

L= int. √((x’(t)²-y’(t)²)

Arc length polar

L= int. √(r²-(r’)²)

Average value

L= 1/b-a int. f(x)

Cart to polar

r = √x²+y² , θ=tan(y/x)

Polar to cart

x=rcosθ , y=rsinθ

ET<

k(b-a)³ /12n²

ES<

k(b-a)^5 /180n^4

1/1-x

Σx^n

e^x

x^n/n!

sinx

Σ (-1)^n x^(2n+1)/(2n+1)!

cosx

Σ (-1)^n x^(2n)/(2n)!

Tan^-1x

Σ (-1)^n x^(2n+1)/2n+1

Ln(1-x)

(-1)^n-1 (x^n / n)

(1+x)k

Σ (k/n)x^n

Doubled exp

A0 × 2^(t / t0)

Halved exp

A0 × 2^(t0 / t)

Newtons law of cooling

DT/ dt = k(T-T0)

Work for spring

W=int. kxdx = ½ k(b²-a²)

Pumping fluids work

W= int. (weight density)(area)(distance)dy

Remainder for Taylor series

Rn(x)= f^{n+1}(z)(x-c)^n+1 / (n+1)!

N is, 1st term is, z is

N is how many terms there are (highest exponent)

1st term is x-c

Z is between x and c and is the max value of f^{n+1}