Pure Notes

Question 1

What does tanx differentiate to?

Answer 1

Sec²x

Question 2

What is the integral of sec²x?

Answer 2

Tanx+c

Question 3

What does cosecx differentiate to?

Answer 3

-cosecxcotx

Question 4

What is the integral of cosecxcotx?

Answer 4

-cosecx+c

Question 5

What does cotx differentiate to?

Answer 5

-cosec²x

Question 6

What is the integral of cosec²x?

Answer 6

-cotx+c

Question 7

What does secx differentiate to?

Answer 7

Secxtanx

Question 8

What is the integral of secxtanx?

Answer 8

Secx+c

Question 9

What is the integral of 1/x ?

Answer 9

Ln|x|

Question 10

What is the integral of 5e^x?

Answer 10

5e^x (the same)

Question 11

What is the integral of 2/x² ?

Answer 11

-2x^-1
A.k.a -2/x

Question 12

What does 1/sin²x equal?

Answer 12

Cosec²x

Question 13

What are 2 rules for integration?

Answer 13

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

Question 14

What does sin(2x+3) differentiate to?

Answer 14

2cos(2x+3)

Question 15

What does e^(4x+1) differentiate to?

Answer 15

4e^(4x+1)

Question 16

What does tan3x differentiate to?

Answer 16

3sec²3x

Question 17

What is the integral of cos(2x+3)?

Answer 17

½sin(2x+3) + c

Question 18

What is the integral of e^(4x+1)?

Answer 18

¼e^(4x+1) + c

Question 19

What is the integral of sec²3x?

Answer 19

⅓tan3x + c

Question 20

What is the integral of 2/(3x-4)?

Answer 20

⅔ln|3x-4|

Question 21

What is the integral of tan²x?

Answer 21

Tanx - x + c
(Using tan²x ≡ sec²x - 1)

Question 22

What is the integral of sin²x?

Answer 22

½x - ¼sin2x + c
(Using rearranged cos double angle: cos2x ≡ 1 - 2sin²x to get that sin²x ≡ ½ + ½cos2x)

Question 23

What is the integral of cos²x?

Answer 23

¼sin2x + ½x + c
(Using rearranged cos double angle: cos2x ≡ 2cos²x-1 to find that cos²x ≡ ½cos2x + ½)

Question 24

What is the integral of cot²x?

Answer 24

-cotx - x + c
(Using rearranged identity: 1+cot²x≡cosec²x to find that cot²x≡cosec²x-1)

Question 25

What integration rule is used for finding the integral of 2cosx/sinx?

Answer 25

Reverse chain rule. Form 1 Set y=ln|f(x)|, find derivative then adjust constant

Question 26

What integration rule is used for finding the integral of 6(2x+3)(x²+3x)² ?

Answer 26

Reverse chain rule. Form 2 Set y=[f(x)]^(n-1) find derivative then adjust constant

Question 27

When is the chain rule used for integration?

Answer 27

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)

Question 28

What is the integral of e^x/(e^2x+3) ?

Answer 28

½ln|e^2x + 3| +c

Question 29

What is the integral of cosec²2xcot2x ?

Answer 29

-¼(cot2x)² + C Using reverse chain rule form 2

Question 31

How do we select our Substitution when integrating? (3 types)

Answer 31

(...)^n --> inside the bracket is the sub. The denominator of a fraction e^... --> the power is the sub.

Question 32

What must you remember to do when integrating by Substitution?

Answer 32

Change the limits