Unit 4 - The Most Integral Piece

Fundamental Theorem of Calculus

100

 int_a^b f'(x) dx 

f(b) - f(a)

200

 d/dxint_0^x f(t) dt 

f(x) 

300

 d/dxint_0^(x^2) f(t) dt 

f(x)(2x)

400

 d/dxint_0^(x^5) f(t) dt 

f(x)(5x^4)

500

 F(x) = int_2^(x^2) (t^2-2t+5) dt 

(x^4-2x^2+5)(2x) 

or

2x^5-4x^3+10x

 

Indefinite Integrals

100

 int (2x+6) dx 

x^2+6x+c

 

200

 int (3x^2+6x+5) dx 

x^3+3x^2+5x+c

300

 int (5x^5+4x^2) dx 

(5x^6)/6+(4x^3)/3+c

400

 int (7x^5+3x^2+2) dx 

(7x^6)/6+x^3+2x+c

500

int (9x^4+3x^3+6x^2+3+6/x^2) dx

 

(9x^5)/5+(3x^4)/4+(3x^4)/2+3x-3x^-2+c 

or

(9x^5)/5+(3x^4)/4+(3x^4)/2+3x-3/x^2+c 

Definite Integrals

100

int_0^2 (2x+3) dx

10

200

 int_0^2 (x^2+2) dx 

20/3

300

int_(-2)^0 (x^3+4x) dx

8

400

int_0^2 cos^2(\theta) d\theta

 pi 

500

\int _1^2\frac{2x^5-x+3}{x^2}

 -\ln(2\)+9 

U-Substitution

100

   int (2x+1)^2 dx 

((2x+1)^3)/6 + c

200

 int cos(x)/sin(x) dx 

sin^2(x)/2+c

300

 int sin(x)/(1+cos(x)) dx 

−ln|cos(x)+1|+c

400

 int x^3/(x^2-4) dx 

x^2/2+2ln|x^2−4|+c

500

 int_0^2 (x^2+2) dx 

4