Integration

Common Integrals

100

ƒ 1/x dx

What is lnx + c

200

ƒ e^5x-9 dx

(1/5)e^5x-9

300

Why do we add a constant to an indefinite integral?

To account for constant terms can disappear when we take a derivative

400

ƒ(7x+2)³dx

(1/28)(7x+2)⁴

500

ƒ 1/(1+6x)² dx

(-1/(36x+6))+c

Definite Integrals

100

ƒ x^2 dx from 0 to 2

What is 8/3

200

ƒ 5x^4 + 2 dx from 4 to 10

What is 98988

300

What is the significance of a definite integral?

It represents the area under the curve between two points

400

ƒ cosx dx from (π/6 to π/2)

1/2

500

ƒ xsinx dx from (0 to π/3)

(1/2)sqrt(3)-(π/6)

Integration by Substitution/ Parts

100

∫ (sin³x)(cosx)dx

0.25sin⁴x+c

200

∫  lnx dx

xlnx-x+c

300

ƒ(x²)(e^x)dx

e^x(x²-2x+2)+c

400

ƒ e^xsin(x) dx

(0.5e^x)(sinx-cosx)

500

ƒ(sin³x)(cos⁴x)dx

(1/7)cos⁷x-(1/5)cos⁵x+c

Applications

100

Find the area bounded by x=1, x=3, f(x)=x³+11 and g(x)=3

36

200

Find the area between f(x)=sqrt(x-7), the y-axis, and the lines y=0 and y=1

22/3

300

Find the area bound by f(x)=sqrt(x) and g(x)=x²

1/3

400

What is the volume obtained by rotating f(x)=cosx about the x axis between x=pi/6 and x=pi/3

(π²/12)

500

The region bound by f(x)=4-x² and g(x)=3 is rotated about the x-axis. What is the volume of the solid generated?

(16π)/5