Integration Techniques

Question 1

What is the formula for Integration by Parts?

Answer 1

∫u dv = uv - ∫v du

Question 2

Fill in the blank: In the Integration by Parts formula, ‘u’ is typically chosen to be the function that is _______.

Answer 2

easier to differentiate

Question 3

Fill in the blank: The derivative of ln(x) is ____.

Answer 3

1/x

Question 4

What is the integral of ln(x) dx?

Answer 4

x ln(x) - x + C

Question 5

True or False: ln(ab) = ln(a) + ln(b) for any positive a and b.

Answer 5

True

Question 6

Fill in the blank: The derivative of e^x is ____.

Answer 6

e^x

Question 7

True or False: ln(x^n) = n ln(x).

Answer 7

True

Question 8

What is the chain rule for derivatives?

Answer 8

If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).

Question 9

What is the integral of 1/x dx?

Answer 9

ln|x| + C

Question 10

What is the relationship between ln and exponential functions?

Answer 10

ln(e^x) = x.

Question 11

What is the derivative of x ln(x)?

Answer 11

ln(x) + 1

Question 12

What is the integral of a constant k?

Answer 12

kx + C

Question 13

Fill in the blank: The integral of e^(kx) dx is ____.

Answer 13

(1/k)e^(kx) + C

Question 14

Which identity helps in simplifying integrals involving sin²(x) and cos²(x)?

Answer 14

Pythagorean identity: sin²(x) + cos²(x) = 1

Question 15

What trigonometric substitution is used for integrals involving √(x² - a²)? And what is the domain ?

Answer 15

x = a sec(θ)

0<theta<pi/2

Or

Pi<theta <3pi\2

Question 16

What substitution can be used for the integral ∫ √(a² - x²) dx? And for what domain?

Answer 16

x = a sin(θ)

-Pi/2<theta<pi/2

Question 17

What is the integral of tan²(x) dx?

Answer 17

tan(x) - x + C

Question 18

What substitution is useful for integrating functions involving √(x² + a²)? And what is the domain ?

Answer 18

x = a tan(θ)

-pi/2<theta<pi/2

Question 19

What is the integral of (1/sqrt(1-x²)) dx?

Answer 19

arcsin(x) + C

Question 20

1-sin^2=

Answer 20

Cos^2

Question 21

1+tan^2=

Answer 21

Sec^2

Question 22

Sec^2 , tan^2 and 1 . How are they related

Answer 22

1 + tan^2(x) = sec^2(x )

Juan + a tan = sexy summer dad

Question 23

Sin^2 =

Answer 23

1/2(1-cos2x)

Question 24

Cos^2 =

Answer 24

1/2(1+cos 2x)

Question 25

Csc^2, cot^2 and 1. How are they related

Answer 25

Csc^2 - cot^2 = 1 Cornerl seccant mines cots

Question 26

Is ln(n) less than or greater than n?

Answer 26

Less than

Question 27

For trig integrals of sin and cos: if the power of cos is odd what do we do

Answer 27

Save 1 cos factor Express the other cos factors in terms of Sin Use u sub with u=sin

Question 28

For trig integrals of sin and cos: if the power of sin is odd then what

Answer 28

Save one sin factor Express other factors of sin in terms of cos Use u sub with u = cosx

Question 29

For trig integrals of sin and cos: if the power of both cos and sin is even then what

Answer 29

Use half angle identities

Question 30

Sinxcosx= ?

Answer 30

1/2 sin(2x)

Question 31

For trig integrals of tanx and secx: if the power of sec is even

Answer 31

Save one factor of SEC^2 Express the other sec terms in tan Use u sub where u = tan x

Question 32

For trig integrals of tanx and secx: if the power of tan is odd

Answer 32

Save one factor of sectan Express the rest in sec Use u sub where u=secx

Question 33

How would you simplify down (6x^2 + 7x +1) and describe the process

Answer 33

(6x+1)(x+1) The first variable in each bracket should multiply to the first term The second variable in each bracket should multiply to the 3rd term

Question 34

How do you calculate favg over an interval ?

Answer 34

1/(b-a) * Sa->b (f(x)) dx

Question 35

Sin(2x)

Answer 35

2sinxcosx

Question 36

For indefinite trig subs after we solve the angle and plug it in what must we not forget to add to the answer

Answer 36

PLUS C

Question 37

When we do a definite integral for trig subs with bounds. WHAT is the first thing we must do after we find the “x=“ statement ?

Answer 37

RESET THE BOUNDS: so if x=sec(theta) and the bounds are 1 and 2 Then the new bounds are : Arcsec (1) and arcsec (2)