Techniques of Integration Concepts

Question 1

How does substitution work?

Answer 1

integral of f(x)dx =
integral of [f(u)(du/dx)]dx =
integral of f(u)du

u = g(x); du/dx = g’(x)

Question 2

How do you substitute by parts?

Answer 2

integral of u dv = u*v - integral of v du
Choose and find u and dv
then find v and du

Question 3

Integrating sin^p(x)cos^q(x)

Answer 3

(a) if p is odd, let u = cos(x)
(b) if q is odd, let u = sin(x)
(c) both even, use double angle formulas

Question 4

Integrating tan^m(x)sec^n(x)

Answer 4

(a) m is odd, let u = sec(x)
(b) n is odd, let u = tan(x)

Question 5

Three methods of trig. sub

Answer 5

1. sqrt(a^2-x^2), try x = a*sin(Θ)
2. sqrt(a^2+x^2), try x = a*tan(Θ)
3. sqrt(x^2-a^2), try x = a*sec(Θ)

Question 6

sinh(x) = ?

Answer 6

(e^x-e^(-x))/2

Question 7

(sinh(x))’ = ?

Answer 7

cosh(x)

Question 8

integral of sinh(x) = ?

Answer 8

cosh(x) + C

Question 9

cosh(x) = ?

Answer 9

(e^x+e^(-x))/2

Question 10

(cosh(x))’ = ?

Answer 10

sinh(x)

Question 11

inegral of cosh(x) = ?

Answer 11

sinh(x) + C

Question 12

tanh(x) = ?

Answer 12

sinh(x)/cosh(x)

Question 13

(tanh(x))’ = ?

Answer 13

sech^2(x)

Question 14

integral of sech^2(x) = ?

Answer 14

tanh(x) + C

Question 15

partial fractions for non-repeating linear factors? (ax+b)

Answer 15

term in decomposition: A/(ax+b)

Question 16

partial fractions for repeated linear factors? (ax+b)^k

Answer 16

A_1/(ax+b) + A_2/(ax+b)^2 + … A_k/(ax+b)^k

Question 17

partial fractions for quadratic factors?
ax^2 + bx + c

Answer 17

(Ax + B)/(ax^2+bx+c)

Question 18

integral f(x)dx from a to ∞ = ?

Answer 18

= lim_(b->∞) for integral f(x)dx from a to b

Question 19

integral f(x)dx from -∞ to b = ?

Answer 19

= lim_(a-> -∞) for integral f(x)dx from a to b

Question 20

integral f(x)dx from a to b = ?
if f(x) is undefined at b

Answer 20

lim_(t->b) for integral f(x) dx from a to t

Question 21

integral f(x)dx from a to b = ?
if f(x) is undefined at c

Answer 21

integral f(x)dx from a to c
+ integral f(x)dx from c to b

Question 22

Numerical integration using trapezoidal method

Answer 22

(a) ∆x = (b-a)/n
(b) A = (∆x/2)(f(a) + 2f(x_1) + 2f(x_2) +…+ f(b))

Question 23

Numerical integration using Simpson’s method

Answer 23

(a) ∆x = (b-a)/n
(b) A = (∆x/3)(f(a) + 4f(x1) + 2f(x2) + 4f(x3) +…+ f(b))