Unit 1 - Calc 2

Order of picking u for integration by parts

LIATE - Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential

cos(Ax)cos(Bx) can be rewritten as:

½ (cos(Ax+Bx)+cos(Ax-Bx))

sin(Ax)sin(Bx) can be rewritten as:

½ (cos(Ax-Bx)-cos(Ax+Bx))

sin(Ax)cos(Bx) can be rewritten as:

½ (sin(Ax-Bx)-sin(Ax+Bx))

integral of f(bx) dx =

1/b F(bx) + C

Cos²x=

1-sin²x

Sin²x=

1-cos²x

When sinx or cosx has an odd power

choose the one with the even power for the u (doesn’t matter which is u when both are odd)

If sinx and cosx both have even powers

Rewrite cos²x as (1+cos(2x))/2 and sin²x as (1-cos(2x))/2

When there are two even powers on sinx and cosx, cos²x can be rewritten as

(1+cos(2x))/2

when there are two even powers on sinx and cosx, sin²x can be rewritten as

(1-cos(2x))/2

tan²x=

sec²x-1

sec²x=

tan²x+1

For tans and secs, when there is an ____ power on tangent, choose ____ for u

odd power on tangent, u = secx

For tans and secs, when there is an ____ power on secant, choose ____ for u

even power on secant, u= tanx

csc²x=

(1+cot²x)

cot²x=

(csc²x -1)

integral of tanx dx =

ln|secx|+C, and -ln|cosx|+C

integral of secx dx =

ln|tanx+secx|+C

integral of cotx dx =

ln|sinx|+C

integral of cscx dx =

ln|cscx-cotx|+C

Trig sub: if a²-x²

choose asin(theta) for x

Trig sub: if x²-a²

choose asec(theta) for x

Trig sub: if x²+a²

choose atan(theta)

ln A/B =

ln A - ln B

complete the square:

divide by two and square. Add and subtract same number.

PFD- term in denominator: ax+b

term in PFD: (A/(ax+b))

PFD- term in denominator: ax²+bx+c (irreducible)

term in PFD: (Ax+B)/(ax²+bx+c)

PFD- term in denominator: (ax+b)²

term in PFD: (A/(ax+b)) + (B/(ax+b)²)

PFD- term in denominator: (ax²+bx+c)² (irreducible)

term in PFD: (Ax+B)/(ax²+bx+c) + (Cx+D)/(ax²+bx+c)²)

integral of 1/(x²+a²) dx =

(1/a) arctan(x/a)+C

If lnx appears in an integral with a polynomial

CHOOSE lnx for u for IBP

Type 1 improper integrals (infinite integrals) are convergent when

p>1

Type 1 improper integrals (infinite integrals) are divergent when

p<=1

Type 2 improper integrals (unbounded integrals) are convergent when

p<1

Type 2 improper integrals (unbounded integrals) are divergent when

p>=1

integral of secxtanx=

secx

integral of cscxcotx=

-cscx

integral of sec²x=

tanx

integral of csc²x=

=cotx