calc 1 teig integrals

∫ sec^2 x dx

tan x + C

∫ sec x dx

ln|sec x + tan x| + C

Integral strategy for ∫ sin^m x cos^n x dx when there is an Odd # of sines

u = cos x, du = -sin x dx, use identity sin^2 x = 1 - cos^2 x

integral of tan x dx

ln |sec x| + c

ln |cos x| + c

Integral strategy for ∫ sin^m x cos^n x dx when there is an Odd # of cosines

u = sin x, du = cos x dx, use identity cos^2 x = 1 - sin^2 x

Integral strategy for ∫ sin^m x cos^n x dx when Both are Even

Use half-angle identities: sin^2 θ = 1/2 - 1/2 cos 2θ and cos^2 θ = 1/2 + 1/2 cos 2θ

∫ sin(kx) dx

-1/k cos(kx) + C

∫ cos(kx) dx

1/k sin(kx) + C

Integral strategy for ∫ tan^m x sec^n x dx when there is an Even # of secants

u = tan x, du = sec^2 x dx, change remaining secants to tans using sec^2 θ = tan^2 θ + 1

Integral strategy for ∫ tan^m x sec^n x dx when there is an Odd # of tans

u = sec x, du = sec x tan x dx, change remaining tans to secants using tan^2 θ = sec^2 θ - 1

Integral strategy for ∫ tan^m x sec^n x dx when there is an Odd # of sec and Even # of tan

Use tan^2 θ = sec^2 θ - 1 to change everything to secants. Then use reduction formula: ∫ sec^n x dx = (sec^(n-2) x tan x)/(n-1) + (n-2)/(n-1) ∫ sec^(n-2) x dx

Identity: tan^2 θ + 1 =

sec^2 θ

Identity: tan^2 θ =

sec^2 θ - 1