What are antiderivatives? How are they connected to derivatives? How do we deter
ID: 2869759 • Letter: W
Question
What are antiderivatives? How are they connected to derivatives?
How do we determine an antiderivative? What formulas can we use?
What is an indefinite integral? How is it related to antiderivatives?
Why does the indefinite integral require +c on the end of its solutions? Why is the +c not needed for a definite integral?
Why is (x+5)3 a correct anitiderivative of 3(x+5)2, whereas (2x+5)3 is not a correct antiderivative of 3(2x+5)2 ?
What is the power rule for integration? How do we use it?
What is a definite integral? How is it connected to area under the curve?
How do we find area under the curve using Riemann Sums?
What is the Fundamental Theorem of Calculus? How do we use it to evaluate a definite integral?
How do we solve an integral using substitution?
Explanation / Answer
What are antiderivatives? How are they connected to derivatives?
Answer :
Antiderivatives are the exact opposite of derivatives. They are connected by the fact that "the derivative of an antiderivative of a function is the function itself".
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How do we determine an antiderivative? What formulas can we use?
Answer :
We find antiderivatives by the usage of a bunch of formulae. Formulae include "power rule" , integration by parts, integration by partial fractions, etc
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What is an indefinite integral? How is it related to antiderivatives?
Answer :
An indefinite integral is an integral calculated with no limits of integration.
It is related to antiderivatives by the fact that they mean one and the same thing.
Antiderivative and integral can be used interchangeably
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Why does the indefinite integral require +c on the end of its solutions? Why is the +c not needed for a definite integral?
Answer :
It needs +c at the end because there may be some initial condition of the problem that we are not aware of.
+c is not required for definite because when an integral is limited, we get a definite answer, one without the presence of unknown constants like c
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Why is (x+5)3 a correct anitiderivative of 3(x+5)2, whereas (2x+5)3 is not a correct antiderivative of 3(2x+5)2 ?
Answer :
(x + 5)^3 has x + 5 inside.. When we do a u-substitution, u = x + 5, we get du = dx upon deriving.
So, the power rule can simply be used.
(2x + 5)^3 has 2x + 5 inside. When we do a u-substitution, u = 2x + 5, we get du = 2dx upon deriving, which gives us dx = du/2. So, in this case, since there is du/2, the correct answer would be : (2x + 5)^3 / 2
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What is the power rule for integration? How do we use it?
Answer :
Integral of x^n is : x^(n+1) / (n+1)
We use it for any problem of the form x^n, like x^2 or x^(3/2) or even x^(-1/2)
Integral of x^2 is : x^(2 + 1) / (2 + 1) ---> x^3 / 3
Integral of x^(-1/2) = x^(-1/2 + 1) / (-1/2 + 1) = x^(1/2) / (1/2) = 2x^(1/2)
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What is a definite integral? How is it connected to area under the curve?
Answer :
A D.I is an integral which has limits.
The Definite integral is nothing but the area under the curve of a function.
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How do we find area under the curve using Riemann Sums?
Answer :
We can find it using the Riemann left end, right end or midpoint rules.
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How do we solve an integral using substitution?
Answer :
Lets take an example :
(integral) (2x + 5)^2 * dx
Let u = 2x + 5
du = 2dx
du/2 = dx
So, when we substitute in terms of u , (2x + 5)^2 becomes u^2 and dx becomes du/2
So, the integral becomes :
(integral) u^2 * du/2
Taking constant 1/2 outta the integral :
(1/2) * (intergal) u^2*du
Using power rule :
(1/2) * u^3/3 + C
(1/6)u^3 + C
Plug back u :
(1/6)(2x + 5)^3 + C
This is how it is used