Calculus 2 - Trig Identities

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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