Math 19B Midterm

Fundamental Theorem of Calculus: Part I

on [a, b], ∫f(x)dx = F(b)-F(a)

Fundamental Theorem of Calculus: Part II

on [u(x), v(x)] d/dx∫f(t)dt = f(u(x))u'(x)-f(v(x))v'(x)

∫sin(x)dx

-cos(x)+C

∫cos(x)dx

sin(x)+C

∫sec²(x)dx

tan(x)+C

∫sec(x)tan(x)dx

sec(x)+C

∫1/xdx

ln|x|+C

∫xⁿdx

(1/n+1)xⁿ⁺¹+C

∫(e^x)dx

(e^x)+C

∫sin(nx)dx

-(1/n)cos(nx)+C

∫1/√(1-x²)dx

sin⁻¹(x)+C

∫1/(1+x²)dx

tan⁻¹(x)+C

∫1/(|x|√(x²-1))dx

sec⁻¹(x)+C

∫tan(x)dx

-ln|cos(x)|+C

∫sec(x)dx

ln|sec(x)+tan(x)|+C

∫ln(x)dx

xln(x)-x+C

∫(b^x)dx

(b^x)/ln(b)+C

Integration by Parts

∫u(x)v'(x)dx = u(x)v(x)-∫u'(x)v(x)dx

Reduction Formula: ∫sinⁿ(x)dx

-(1/n)sinⁿ⁻¹(x)cos(x)+(n-1/n)∫sinⁿ⁻²(x)dx

Reduction Formula: ∫cosⁿ(x)dx

(1/n)cosⁿ⁻¹(x)sin(x)+(n-1/n)∫cosⁿ⁻²(x)dx

Substitution: √(a²-x²)

x=asinθ, dx=acosθdθ, √(a²-x²)=acosθ

Substitution: √(x²+a²)

x=atanθ, dx=asec²θdθ, √(x²+a²)=asecθ

Substitution: √(x²-a²)

x=asecθ, dx=asecθtanθdθ, √(x²-a²)=atanθ

∫sin^m(x)cos^n(x)dx, at least one of m, n is odd

if m is odd, u=cos(x)

if n is odd, u=sin(x)

∫sin^m(x)cos^n(x)dx, m, n are even

use sin²(x)=½(1-cos(2x)) and/or cos²(x)=½(1+cos(2x))

∫sec^m(x)tan^n(x)dx, m is even

u=tan(x), du=sec²(x)

∫sec^m(x)tan^n(x)dx, n is odd, m≥1

u=sec(x), du=sec(x)tan(x)

Disk Method

π∫(f(x))²dx

Washer Method

π∫(f(x))²-(g(x))²dx for f(x)>g(x)

Shell Method

2π∫x(f(x))dx

Volume of a Cone

πr²(h/3)

Volume of a Sphere

(4/3)πr³

Volume of a Square Pyramid

(l×w×h)/3

Volume of a Triangular Pyramid

(1/3)(area×height)

Volume of a Cylinder

πr²h

∫xsin(x)dx

-xcos(x)+sin(x)+C

∫xcos(x)dx

xsin(x)-cos(x)+C

∫x(e^x)dx

(x-1)(e^x)+C