Math Chapter 8 Notes

d/dx sinx = cosx

ʃcosxdx = sinx + C

d/dx cosx = - sinx

ʃsinxdx = - cosx + C

d/dx tanx = sec²x

ʃsec²xdx = tanx + C

d/dx cotx = -csc²x

ʃcsc²xdx = -cotx + C

d/dx secx = secxtanx

ʃsecxtanx = secx + C

d/dx cscx = - cscxcotx

ʃcscxcotx = - cscx + C

d/dx e^x = e^x

ʃe^xdx = e^x +C

d/dx ln|x| = 1/x

ʃ1/x dx = ʃdx/x = ln |x| + C

d/dx arcsinx = 1/√1-x²

ʃ1/√1-x² dx = ʃdx/√1-x² = arcsinx + C

d/dx arctanx = 1/1+x²

ʃ1/1+x² dx = ʃdx/1+x² = arctanx + C

d/dx arcsec |x| = 1/x√x²-1

ʃ1/x√x²-1 dx = ʃdx/x√x²-1 = arcsec |x| + C

ʃsin(ax)dx

-1/a cos(ax) + C

ʃcos(ax)dx

1/a sin(ax) + C

ʃtan(ax)dx

1/a ln|sec(ax) + tan (ax)| + C

ʃcsc(ax) dx

-1/a ln|csc(ax) + cot(ax)| + C

Integration by parts equation

ʃudv = uv - ʃvdu

(lnx will be u, harder one will be dv)

Integrals with odd powers: sin²x

1-cos²x

Integrals with odd powers: cos²x

1-sin²x

Integrals with even powers: sin²x

1-cos2x / 2

Integrals with even powers: cos²x

1+cos2x / 2

Trig substitution a² - x²

x = asin⊖

Known identity: a² - a² sin²⊖ = a²cos²⊖

Trig substitution: a² + x²

x = atan⊖

Known identity: a² + a²tan²⊖ = a²sec²⊖

Trig substitution: x² - a²

x = asec⊖

Known Identity: a²sec²⊖ - a² = a²tan²⊖

Pythagorean identity: sin²⊖

1- cos²⊖

Pythagorean Identity: cos²⊖

1 - sin²⊖

Pythagorean Identity: sec²⊖

1+ tan²⊖

If p > 1

ʃ∞-1 1/x^p dx = 1/1-p

If p < or = 1

ʃ∞-1 1/x^p dx diverges