5.6 Integrals Involving Exponential and Logarithmic Functions

Find the antiderivative of the function using substitution x²e^-²x^3

(-1/6)e^-²2x3 + C 1. let u = -2x^3 du = -6x²dx (-1/6) du = -6x²dx(-1/6) (-1/6) du= x²dx 2. e^u • du (-1/6)e^-2x^3 + C

Example: Fruit Fly Population Growth: Evaluating the Integral of an Exponential Function and Applying the Net Change Theorem Suppose a population of fruit flies increases at a rate of g(t) = 2e^0.02t, in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?

*Remember the net change theorem is when we find the integral and evaluate it at the upper (b) and lower limit (a) and find the difference: F(b) - F(a)

Example: Growth of Bacteria in a Culture Suppose the rate of growth of bacteria in a Petri dish is given by q(t) = 3^t, where t is given in hours and q(t) is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function Q(t) that gives the number of bacteria in the Petri dish at any time t. How many bacteria are in the dish after two hours?

*Remember the rules for exponential functions involving constants: a^xdx = a^x/lna +C 1. 3^t dx = 3^t/ln3 + C 2. Evaluate at t = 0: 3^0 = 1, so 1/ln3 + C 3. Solve for C by isolating it: 10 - 1/ln3 = C C = 9.090 Now Q(t) = 3^t/ln3 + 9.090 4. Now find how many bacteria are in the dish when t = 2: Q(2) = 3²/ln3 + 9.090 = 17.282

Example: Finding an Antiderivative of a Rational Function Find the antiderivative of (2x³+3x)/(x^4+3x²)

*remember the rule for integrations of logarithmic functions* ∫x^-1 dx = |x| + C

Suppose the rate of growth of the fly population is given by g(t) = e^0.01t, and the initial fly population is 100 flies. How many flies are in the population after 15 days?

1. G(15) = G(0) ∫ e^0.01t dt as 0 approaches 15 u = e^0.01t du = 0.01 dt 1 du/0.01 = 0.01/0.01 dt = (1/0.01)e^0.01t evaluate at upper limit of 15 and lower limit of 0 2. 100 (initial fly population) + [100e^0.01t] evaluated at upper limit of 15 and lower limit of 0: find their difference to use the net theorem 3. 100 + {116.183 - 100] = 116.18 After 15 days there are 116 flies

Suppose the rate of bacteria in a Petri dish given by q(t) = 2^t, where t is given in hours and q(t) is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function Q(t) that gives the number of bacteria in the Petri dish at any time t. How many bacteria are in the dish after 3 hours?

1. q(t) = 2t Q(t) = 2^t dx a^x dx = a^x/lna + C = 2^t/ln2 + C Q(t) = 2^t/ln2 + C Q(0) = 10 = 1/ln2 + C because 2^0/ln2 + C = 1/ln2 + C 2. Solve for C: 10 - 1/ln2 = C C = 8.557 3. Find Q(3): Q(3) = 2^3/ln2 + 8.557 = 20.099 = 20,099 bacteria in the dish after 3 hours

Evaluate the definite integral of e²x dx as 0 approaches 2

1. u = 2x du = 2 dx 1/2 du = 2 dx (1/2) 1/2 du = dx 2. (1/2) e^u dx 3. upper limit for u at x = 2: 2(2) = 4 lower limit for u at x = 0: 2(0) = 0 Limit for u = (0,4) 4. (1/2) e^4 - e^0 evaluate at the upper and lower limits for u = 1/2 (e^4 - 1) e^0 = 1

Find the antiderivative of e^x(3e^x - 2)²

1. u = 3e^x -2 du = 3e^x dx 1/3 du = 3e^x dx (1/3) 1/3 du = e^x dx 2. (1/3) • u² • du = (1/3) • u³/3 • du. use power rule = (1/9) (3e^x -2)^3 + C. plug in the og values

Evaluate the definite integral using substitution ∫ (1/x³)e^4x-²dx

1. u = 4x-² du = -8x-³ (-1/8) du = (-8/x³)dx(-1/8) (-1/8) du = (1/x³) 2. (-1/8) ∫ e^u dx 3. Find the upper and lower limit for u upper limit @ x = 2: 4(2)-² = 1 lower limit @ x = 1: 4(1)-² = 4 (1,4) is the interval for u 4. Evaluate at the upper and lower limits using the net theorem e^4 - e^1 = e^4 - e *e^1 = 1 5. (1/8)(e^4-e)

Example: Use Substitution with an Exponential Function Use substitution to evaluate the indefinite integral ∫ 3x²e²x³dx

Notice that we don't use the power rule here because u is not in parenthesis raised to an exponent

Integrals of Exponential Functions

a is a constant *we cannot use the power rule for the exponent on e*

Evaluating a Definite Integral Find the definite integral of (sin x)/(1 + cos x) dx from 0 to π/2

Toward the end of the problem remember the quotient property of logarithms logb(x) - logb(y) = logb(x/y), so we end up dividing 2/1 = 2.

Find the antiderivative of log3x

Use the rule on integration formulas involving logarithmic functions ∫logax dx = (x/lna)(ln x-1) + C