Indefinite Integrals. Objectives Students will be able to Calculate an indefinite integral....
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Transcript of Indefinite Integrals. Objectives Students will be able to Calculate an indefinite integral....
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
)(