Extra Integration Techiniques

Question 1

Integration by Parts

Answer 1

∫udv = uv - ∫vdu

Question 2

Integrals in the form: ∫(x^n)(e^ax)dx, ∫(x^n)(sin(ax))dx, or ∫(x^n)(cos(ax))dx, ….

Answer 2

u = (x^n)
dv = (e^ax)dx or sin(ax)dx or cos(ax)dx

Question 3

Integrals in the form: ∫(x^n)(lnx)dx, ∫(x^n)(arcsin(ax))dx, or ∫(x^n)(arctan(ax))dx

Answer 3

u = lnx or arcsin(ax) or arctan(ax)
dv = (x^n)dx

Question 4

Integrals in the form: ∫(e^ax)(sin(bx))dx or ∫(e^ax)(cos(bx))dx

Answer 4

u = sin(bx) or cos(bx)
dv = (e^ax)dx

Question 5

Guidelines for Integration by Parts

Answer 5

1. Try letting (dv) be the most complicated portion of the integrand that fits a basic integration rule. Then (u) will be the factor(s) of the integrand
2. Try letting (u) be the portion of the integrand whose derivative is a function simpler than (u). Then (dv) will be the remaining factor(s) of the integrand.

Question 6

Integrals involving √((a^2) - (u^2))

Answer 6

u = asin(x)

If (-π/2≤x≤π/2), then u = acos(x)

Question 7

Integrals involving √((a^2) + (u^2))

Answer 7

u = atan(x)

If (-π/2

Question 8

Intergrals involving √((u^2) - (a^2))

Answer 8

u = asec(x)
u = atan(x) if u>a, where 0≤x≤π/2
u = -atan(x) if u

Question 9

x^2 needs to (trig sub)…

Answer 9

always have a coefficient of -1 or 1