Integration

Question 1

What is an indefinite integral?

Answer 1

The inverse of differentiation

Question 2

What is a definite integral?

Answer 2

The area under a curve

Question 3

How are integrals represented when limits involve an infinity?

Answer 3

Integral between ∞ and a for f(x) dx = lim as b->∞ integral between b and a for f(x) dx

Question 4

What is the integral of e^(-λx) between 0 and ∞ equal to?

Given that λ > 0

Answer 4

1/λ

Question 5

What should be done if an integrand f(x) contains a finite number of discontinuities in the integration range?

Answer 5

Break up the range into regions over which f(x) is continuous

Question 6

What are the ten integration tricks?

Answer 6

Inspection
Substitution
Trig substitutions
Hyperbolic substitutions
Partial fractions
Complete the square
Integration by parts
Trig identities
Complex numbers
Symmetry

Question 7

Integrals involving sqrt(a^2 - x^2), what should be subbed in for x?

Answer 7

x = asin(θ)

Question 8

Integrals involving a^2 + x^2, what should be subbed in for x?

Answer 8

x = atan(θ)

Question 9

Integrals involving sqrt(x^2 + a^2), what should be subbed in for x?

Answer 9

x = asinh(y)

Question 10

Integrals involving sqrt(x^2 - a^2), what should be subbed in for x?

Answer 10

x = acosh(y)

Question 11

Integrals involving a^2 - x^2, what should be subbed in for x when |x| < or > a?

Answer 11

x = atanh(y) (|x| < a)
x = acoth(y) (|x| > a)

Question 12

When is the complete the square trick used?

Answer 12

Integrands of the form 1/sqrt(Q(x)) or 1/Q(x)
Where Q(x) is quadratic

Question 13

Integration by parts formula

Answer 13

∫fg’ dx = fg - ∫gf’ dx

Question 14

Value of integral between infinity and 0 for (x^n e^-x)

Answer 14

n! = Γ(n+1)

Question 15

Stirling’s approximation

Answer 15

ln(n!) roughly equals n ln(n) - n

Question 16

A better approximation for n!

Answer 16

n! roughly equals n^n e^-n sqrt(2πn)

Question 17

Schwarz’s inequality

Answer 17

(∫ f(x)g(x) dx)^2 =< (∫ f^2(x) dx)(∫ g^2(x) dx)

all integrands between the same limits

Question 18

Volume element for Cartesian

Answer 18

dV = dx dy dz

Question 19

Volume element for Cylindrical polars

Answer 19

dV = r dr dΦ dz

Question 20

Volume element for Spherical polars

Answer 20

dV = r^2 sin(θ) dr dθ dΦ

Question 21

What is the value of the ∫ e^(-x^2) dx between +/- infinity?

Answer 21

sqrt(π)