Calculus 2 - Trig Identities

dy/dx sin x

= cos x

dy/dx cos x

= -sin x

dy/dx tan x

= sec^2(x)

dy/dx cot x

= -csc^2(x)

dy/dx sec x

= sec x tan x

dy/dx csc x

= csc x cot x

∫ sin x dx

= -cos x + C

∫ cos x dx

= sin x + C

∫ tan x dx

= ln |sec x| + C

∫ cot x dx

= ln |sin x| + C

∫ sec x dx

= ln |sec x + tan x| + C

∫ csc x dx

= ln |csc x + cot x| + C

∫ 1/x dx

= ln |x| + C

cos(2x)

= cos^2(x) - sin^2(x)

cos^2(x) + sin^2(x)

= 1

sin (2x)

= 2 sin x cos x

cos^2(x)

= 1 - sin^2(x)

cos^2(x)

= (1+cos2x) / 2

sin^2(x)

= 1 - cos^2(x)

sin^2(x)

= 1 - cos (2x) / 2

∫ e^kx dx

= e^kx / k + C

∫ sec^2(x)

= tan x + C

∫ csc^2(x) dx

= -cot x + C

∫ csc x cot x dx

= -csc x + C

1 + cot^2(x)

= csc^2(x)

tan^2(x) + 1

= sec^2(x)

What trig substitution does √(a^2 - x^2) belong with ? What is it's identity?

x = a sin θ; 1 - sin^2(x) = cos^2(x)

What trig substitution does √(a^2 + x^2) belong with ? What is it's identity?

x = a tan θ; 1 + tan^2(x) = sec^2(x)

What trig substitution does √(x^2 - a^2) belong with ? What is it's identity?

x = a sec θ; sec^2(x) - 1 = tan^2(x)

How do you tell if a function is convergent?

The answer to the integral is a finite number.

How can you tell if a function is divergent?

The answer to the integral is an infinite number (infinity).

What is the integration by parts formula?

∫u dv = uv - ∫v du