Evaluate the indefinite integral. Remember the + C.
The simplest antiderivative of this is x-1, or , so .
To find the antiderivative, take the antiderivative of each term:
Determine if the integral is a definite integral or an indefinite integral.
This is a definite integral, because it has limits of integration:
This is an indefinite integral because the integral sign has no limits of integration.
Assume the function f ( x ) can be integrated over any interval. Determine if the statement is true or false. Explain your reasoning.
is a number.
True. is definite integral, so it's a number.
is a family of functions.
False. is a definite integral. It's just one number, not a family of functions.
False. is not just one number - it's the family of all functions with derivative <em>f</em> ( <em>x</em> ).
is a function.
False. is a collection of infinitely many functions. It's not just one single function.
True. is the family of functions with derivative f(x).
False. and can't be equal, because one is a number and the other is a collection of infinitely many functions. The equation
doesn't make any sense.
Determine if the answer is correct.
Take the derivative of the answer:
We got the integrand, so the answer 4(x2 + 4) + C is correct.
This is not equal to the original integrand, so the answer (2x + 3)5 + C is not correct.
for all x > 0
This is the original integrand, so the answer x ln x – x + C is correct.
This is not equal to the integrand, so the answer
is not correct.
for all x for which (x + 2) and (3x – 7) are positive.
Since we got the original integrand back, was the correct answer.
Make it rain.