Indefinite Integrals

Objectives

Students will be able to• Calculate an indefinite integral.• Calculate a definite integral.

Definition

The symbol ∫is the integral sign; f(x) is the integrand; x is the variable of integration; and C is the constant of integration.

)()(where)()( xfxFCxFdxxf

Important Integrals

)1(1

1 1 aCxa

dxx aa

Cxdxx

ln1

)0(1

aCea

dxe axax

)1and0(ln1

aaCaa

dxa xx

General Rules of Integration

dxxfadxxaf )()(

dxxgdxxfdxxgxf )()()()(

where a is a constant

Example 1

Evaluate the indefinite integral

dy9

Example 2

Evaluate the indefinite integral

dxxx )365( 2

Example 3

Evaluate the indefinite integral

dxxxx )34( 42

Example 4

Evaluate the indefinite integral

dxx

x3

1

Example 5

Evaluate the indefinite integral

dxex

x4.039

Example 6

Evaluate the indefinite integral

dxx3

Example 7

Evaluate the indefinite integral

dyy 2)12(

Example 8Under certain conditions, the number of

diseased cells N(t) at time t increases at a rate

where A is the rate of increase at time 0 (in cells per day) and k is a constant.

a. Suppose A = 60, and at 4 days, the cells are growing at a rate of 300 per day. Find a formula for the number of cells after t days, given that 400 cells are present at t = 0.

b. Use your answer from part a to find the number of cells present after 11 days.

ktAetN )(

Example 9

Suppose

v(0)= –7, and s(0) = 12. Find s(t).

tetta 7245

)(