Pure Notes Flashcards
What does tanx differentiate to?
Sec²x
What is the integral of sec²x?
Tanx+c
What does cosecx differentiate to?
-cosecxcotx
What is the integral of cosecxcotx?
-cosecx+c
What does cotx differentiate to?
-cosec²x
What is the integral of cosec²x?
-cotx+c
What does secx differentiate to?
Secxtanx
What is the integral of secxtanx?
Secx+c
What is the integral of 1/x ?
Ln|x|
What is the integral of 5e^x?
5e^x (the same)
What is the integral of 2/x² ?
-2x^-1
A.k.a -2/x
What does 1/sin²x equal?
Cosec²x
What are 2 rules for integration?
1) ∫k f(x) dx
= k ∫f(x) when k is a constant.
2) swapping limits means change sign.
∫ f(x) dx from ª to b = ∫ - f(x) dx from b to ª
What does sin(2x+3) differentiate to?
2cos(2x+3)
What does e^(4x+1) differentiate to?
4e^(4x+1)
What does tan3x differentiate to?
3sec²3x
What is the integral of cos(2x+3)?
½sin(2x+3) + c
What is the integral of e^(4x+1)?
¼e^(4x+1) + c
What is the integral of sec²3x?
⅓tan3x + c
What is the integral of 2/(3x-4)?
⅔ln|3x-4|
What is the integral of tan²x?
Tanx - x + c
(Using tan²x ≡ sec²x - 1)
What is the integral of sin²x?
½x - ¼sin2x + c
(Using rearranged cos double angle: cos2x ≡ 1 - 2sin²x to get that sin²x ≡ ½ + ½cos2x)
What is the integral of cos²x?
¼sin2x + ½x + c
(Using rearranged cos double angle: cos2x ≡ 2cos²x-1 to find that cos²x ≡ ½cos2x + ½)
What is the integral of cot²x?
-cotx - x + c
(Using rearranged identity: 1+cot²x≡cosec²x to find that cot²x≡cosec²x-1)
What integration rule is used for finding the integral of 2cosx/sinx?
Reverse chain rule. Form 1
Set y=ln|f(x)|, find derivative then adjust constant
What integration rule is used for finding the integral of 6(2x+3)(x²+3x)² ?
Reverse chain rule. Form 2
Set y=[f(x)]^(n-1) find derivative then adjust constant
When is the chain rule used for integration?
When the derivative of the function is also present. Either with the derivative as the numator or as a factor being multiplied (which is also multiplied by k constant)
What is the integral of e^x/(e^2x+3) ?
½ln|e^2x + 3| +c
What is the integral of cosec²2xcot2x ?
-¼(cot2x)² + C
Using reverse chain rule form 2
How do we select our Substitution when integrating? (3 types)
(…)^n –> inside the bracket is the sub.
The denominator of a fraction
e^… –> the power is the sub.
What must you remember to do when integrating by Substitution?
Change the limits
