Study Guide for Exam 2: Integration Techniques

Integration of sin^odd x times cos^n x

Use u = cosx substitution

Integration of sin^m x times cos^odd x

use u = sinx substitution

Integration of tan^m x times sec^even x, even > 0

use u = tanx substitution

Integration of tan^odd x times sec^n x

use u = secx substitution

if tan^m x sec^0 x

Used tan^2 x = sec^2 x - 1 to reduce degree

Double angle formula + half angle formula

Reduces degree for when sin^even x times cos^even x

For ax, when a>1, remember that sin(bx/2) = sqrt(1-cos(bx) / 2) and b must = 2a so that when you divide by 2 you get back to a

so if you have sin^2(x) it becomes (1-cos(bx)) / 2

or cos^2(x) it becomes (1+cos(bx)) / 2

When to use Integration by parts in trig integration

Used for tan^even x sec^odd x

integral of tan^2 x dx

tanx - x + C

Trigonometric substitution

Used for integrals involving square roots.

x = a sin θ, dx = a cos θ

Substitution for √(a² - x²) integrals, then it = a cos θ

x = a tan θ, dx = a sec θ

Substitution for √(a² + x²) integrals, then it = a sec θ

x = a sec θ, dx = a tan θ sec θ

Substitution for √(x² - a²) integrals, then it = a tan θ

Length of a curve y=x^2 / 2 over the interval [1, sqrt(3)]

the arc length is the integral of sqrt(1+dx^2)dx = integral(1->sqrt(3)) sqrt(1+x^2)dx

do a trig sub using x = tan(theta), dx = sec^2(theta)dtheta, sqrt(1+x^2) = sec(theta)

this becomes the integral(pi/4 -> pi/3) sec^3(theta)dtheta

answer: sqrt(3) + 1/2 ln(2+sqrt(3)) - sqrt(2)/2 - 1/2 ln(sqrt(2) + 1)

Convergence of improper integrals

Determines if an integral converges or diverges.

Divergent integral

Integral that does not approach a finite limit.

Telescoping series

Series where most terms cancel out except a_1 and a_n+1

integral (0 -> ∞) (e^x) / (e^2x + 1) dx

pi / 4

integral (e^5 -> ∞) (1/x[ln(x)]^2) dx

1/5

Compute integral of secxdx using u = sinx and a partial fraction.

This should match ln|secx + tanx| + c

integral sec x dx = integral (1/cos^2 x)(cosx)dx = integral (1/1-sin^2 x)(cosx)dx

u = sinx, du = cosxdx

integral (1/1-u^2)du = integral du/(1-u)(1+u) = A/(1-u) + B/(1+u)

A(1+u) + B(1-u), u = +-1

A = 1/2, B = 1/2

integral 1/2(1-u) + 1/2(1+u) du

ln|1 - u|/2 + ln|1 + u|/2 +c

ln|(1+sinx)(1-sinx)|/2 + c

1/2 ln|cos^2 x| + c

ln|cosx| + c, ln|cosx| = -ln|secx| = ln(1/secx), ln(sec + tan) = ln(sec^2 x), integral sec^2 xdx = secx+tanx+c

ln |sec x + tan x| + c

integral of tan x dx (most common)

ln |sec x| + C

integral (0 -> 2) (1/(x-1)^2) dx

divergent = ∞

integral (0 -> ∞) (e^(2x)/(e^(2x)+1) dx

divergent = ∞

integral (0 -> 9) (1/(x-1)) dx

divergent = -∞

integral (0 -> 9) [1/cube root(x-1)] dx

improper integral → 9/2

integral (-∞ -> ∞) x dx

divergent = ∞

integral (0 -> ∞) (xe^(-x)) dx

1

integral (-∞ -> 0) [xe^(-x²)] dx

-1/2

integral (e -> ∞) (1/xlnx) dx

divergent = ∞

integral (0 -> 3pi/2) sin^5 x cos^2 x dx

u=cosx du=-sinxdx

(2cos³x)/3 - (cos³x)/3 - (cos^5x)/5 (0 → 3pi/2)

8/105

integral (0 -> pi/2) cos²(3x) dx

half angle formula

pi/4

integral (0 -> pi/4) sin^5 x dx

u=cosx du=-sinxdx

pi/8 - 1/4

integral (0 -> pi/3) tan^2 x sec^4 x dx

split one sec²x into 1-tan²x then u=tanx du=sec²xdx

14sqrt(3))/5

integral (0 -> pi/4) tanx sec^5 x dx

u=secx du=secxtanxdx

[64 - sqrt(2)]/5 sqrt(2)

integral (0 -> pi/3) tan^3 x dx

use sec²x-1 in place of tanx then u=secx du=secxtanx

-ln(2) -3/2

integral (1 -> sqrt(2)) x^3(sqrt(x^2 - 1)) dx

x=sec(theta) dx=sec(theta)tan(theta)dtheta

8/15

integral (1 -> sqrt(3)) 1 / x^2(sqrt(x^2 + 1)) dx

x=tan(theta) dx=sec²(theta)

sqrt(2) - 2/(sqrt(3))

integral (-3 ->3) 1 / sqrt(x^2 + 9) dx

x=3tan(theta) dx=3sec²(theta)dtheta

ln(1 + sqrt(2)) - ln(sqrt(2) - 1)

integral (0 -> 7/sqrt(2)) x^2/(sqrt(49-x^2)) dx

x=7sin(theta) dx=7cos(theta)dtheta

49pi/8 - 49/4

integral (0 -> 1/2) x^3/(1-x^2)^2 dx

ln(3/4)/2 + 1/6

sum(n=1 -> ∞) (3/[n^2 + 5n + 6])

becomes 3/(n+2)(n+3)

A/n+2 + B/n+3 = A(n+3) + B(n+2), n=-2 | n=-3

A=3, -B=3, B=-3

a₁=3/3 - ¾ first 2 terms won’t cancel out bc if its A/n+k + B/n+k+1 the first k terms won’t cancel

1

sum(n=1 -> ∞) (2/[n^2 + 2n])

3/2