Indefinite Integral Jeopardy

U-substitution

100

∫sec²x√(tanx)dx

(2/3)(tanx^3/2) + c

200

∫tan(4x+2)dx

-(1/4)ln｜cos(4x+2)｜ + c

300

∫(ln⁶x)dx/x

(1/7)(ln⁷x) + c

400

∫(sin(2t+1))/(cos²(2t+1)) + c

(1/2)sec(2t+1) + c

500

∫xdx/(x²+1)

(1/2)ln(x²+1) + c

Integration By Parts

100

∫3t(e^2t)dt

(3/2)te^2t-(3/4)e^2t + c 

200

∫ylnydy

(y²/2)lny - (y²/4) + c

300

∫e^xsinxdx

(e^x/2)(sinx-cosx) + c 

400

∫x³e^-2xdx

e^-2x(-x³/2 - 3x²/4 - 3x/4 - 3/8) + c

500

∫xsin5xdx

-(x/5)cos5x + (1/25)sin5x + c 

Determine "u"

100

∫ xcos(2x²)dx

u = 2x²

200

∫sec(2x)tan(2x)dx

u = 2x

300

∫(9r²)dr/(√ (1-r³))

u = 1 - r³

400

∫dx/(x²+9)

u = x/3

500

∫8(y⁴+4y²+1)²(y³+2y)dy

u = y⁴+4y²+1

Surprise

100

∫dx/₃√(3x+4)

(1/2)(3x+4)^2/3

200

∫x³cosxdx

x³sinx + 3x²cosx - 6xsinx -6cosx + c

300

∫25dx/(x²-25)

(5/2)ln｜(x-5)/(x+5)｜+ c

400

∫e^xsec(e^x)dx

ln｜sec(e^x)+tan(e^x)｜+ c

500

∫x²e^-3xdx

e^-3x(-x²/3 - 2x/9 -2/27) + c

Mathematical Definitions

100

indefinite integral of f with respect to x

∫f(x)dx = F(x) + c

F'(x) = f(x)

200

constant of integration

c, an arbitrary constant

300

integration by parts formula

∫udv = uv - ∫vdu

400

tabular integration

a way to organize calculations of integration by parts 

f(x) and its derivatives   g(x) and its integrals

x²                                  e^x

2x                                  e^x

2                                    e^x

0                                    e^x

x² multiplies diagonally with e^x, followed by 2x and 2. The values are added alternating positive and negative.

500

LIATE

the order of choosing "u"

L. logarithmic functions

I. inverse trig functions

A. algebraic functions

T. trig functions

E. exponential functions

Highest priority (L) to lowest priority (E)