What is Antiderivatives: Definition and 46 Discussions

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acceleration). The discrete equivalent of the notion of antiderivative is antidifference.

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  1. hugo_faurand

    B Calculate the expression of the antiderivative

    Hello everyone ! I've started to work on integral and I wonder if it's possible to calculate the expression of the antiderivative with the expression of the "integrand"1 rather than use a table with the function and its antiderivative. Thank you in advance ! 1( I'm french and I d'ont know the...
  2. M

    MHB Calculating Integrals with Antiderivatives

    Hey! :o Let $a, b \in \mathbb{R}$ with $a<b$ and $f\in C^1([a,b])$. I want to calculate the following integrals using the fundamental theorem of calculus, by finding in each case an antiderivative. $\int_a^b\frac{f'(x)}{f(x)}$ with $0\notin f([a,b])$ $\int_a^bf'(x)f(x)dx$...
  3. dfklajsdfald

    Solve Antiderivative of xe^x | Step-by-Step Guide

    Homework Statement find the anti-derivative of xe ^x so its x to the power of e to the power of x Homework EquationsThe Attempt at a Solution i have 0 idea where to even start. this was a question on my quiz today
  4. L

    Complex Integrals, Antiderivatives, Logarithms

    I've been teaching myself a little bit of Complex Variables this semester, and I had a question concerning complex integrals. If I understand correctly, then if a function f has an antiderivative F , then the line integral \int_C f(z) dz is path independent and always evaluates to F(z_1)...
  5. P

    Antiderivatives and the fundamental theorem

    I know that according to the first fundamental theorem of calculus: $$\frac{d}{dx} \int_a^x f(t) dt = f(x)$$ I also know that if ##F## is an antiderivative of ##f##, then the most general antiderivative is obtained by adding a constant. My question is, can every single antiderivative of ##f## be...
  6. S

    Integration by parts, just need a small hand

    Homework Statement I'm going to cut from the initial part of the problem, which I am confident is good to go, and cut straight to the antiderivatives. Homework Equations All antiderivatives are to be integrated on the interval from 0 to π/18 (I1) = -1/9 cos 9x - (I2) (-2/27 * cos3(9x)) + (I3)...
  7. thegreengineer

    Integral calculus: integral variable substitution confusion

    Recently I started seeing integral calculus and right now we are covering the topic of the antiderivative. At first sign it was not very difficult, until we started seeing integral variable substitution. The problem starts right here: Let's suppose that we have a function like this: \int...
  8. A

    Just started Antiderivatives Help?

    Homework Statement F[/B]ind the Antiderivative of: (x^3-1)/(x-1). All is known is the integration formulas (i.e. ∫sinx = -cosx+c) Homework Equations Integration Formulas the most complicated being ∫cscx dx= -ln(cscx+cotx)+c The Attempt at a Solution I tried doing (x^3/x-1) -(1/x-1), but now...
  9. andyrk

    Continuity in Integrals and Antiderivatives

    I was a bit confused by the definition of integrals (both definite and indefinite) and anti-derivatives. The definition for indefinite integrals is- The indefinite integral of a function x with respect to f(x) is another function g(x) whose derivative is f(x). i.e. g'(x) = f(x) ⇒ Indefinite...
  10. L

    Integrals/Non-Elementary Antiderivatives

    Hey everyone, I'm wondering how to solve the following definite integral, \int^\infty_{-\infty}{x^4e^{-x^2}dx}. I know the answer is ##\frac{3 \sqrt{\pi}}{4}##, but I'm not positive how to get there. I understand how to evaluate the definite "Gaussian" integral...
  11. L

    LaTeX Latex skills and asking about antiderivatives

    As a way of practicing my Latex skills and asking about antiderivatives. Suppose we have \int\frac{1}{x^2 + a}dx a is just a constant Now I recognize this to be \int\frac{dx}{x} whose antiderivative is \frac{\ln{|x|}}{dx} + C The question is: In the first function, I could make a...
  12. M

    Do not get Logarithmic antiderivatives

    Dear All, the antiderivative of $$\frac{1}{x}$$ is $$ln \vert x \vert $$. If I then consider the expression $$ (x-a) ln (\vert x-a\vert)-x $$ and compute the first derivative I obtain $$ ln (\vert x-a \vert)$$. If I then consider the equivalent expression $$ (x-a) ln (\vert a -x\vert)-x $$...
  13. F

    Antiderivatives of Logarithmic and Radical Functions: Can They Be Solved?

    Homework Statement \int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dxHomework Equations X.The Attempt at a Solution Wolfram Alpha seem to give no answer.
  14. B

    Understanding Trig Identities and Variations

    Homework Statement I don't see how they're going from sec x cot x to csc x and csc x tan x to sec x The derivative of sec x is sec x tan x not csc x tan x and the derivative of csc x is -csc x cot x and if it's an identity then they didn't care to inform me of that at...
  15. N

    How long does it take for a stone thrown from a bridge to hit the water below?

    Homework Statement A stone is thrown up at 30 m/s from the edge of a bridge 210 m above the river below. How many seconds elapse between toss and splash? Homework Equations The Attempt at a Solution dv/dt=-9.8 v=-9.8t+c2 30=-9.8(0) + c1 c1=30 v=-9.8t+30 h=-4.9t^2+30t+c2...
  16. N

    Computing antiderivatives (integration)

    Homework Statement integrate 4e^(2x)^(1/2) - 1/7e^(-pix) using a guess and check method (haven't learned many rules of integration) Homework Equations The Attempt at a Solution i'm not really sure how to do this integral... i tried 4/(2x)^(1/2)[e^(2x)^(1/2)] using a table of...
  17. N

    Motion problems using antiderivatives

    Homework Statement An object moves in a straight line with velocity v = 6t-3t^2, where v is measured in metres per second. a) how far does the object move in the first sec? b) how far does the object move in the first two seconds? c) the object is back where it started when t=3. how far...
  18. R

    Can substitution help solve this antiderivative problem?

    :confused: Problem: (x^2+2x)/sqrt[x^3+3x^2+1] = 2/3/sqrt[(x+3x)^2+1] is this the answer?:confused: i don't knw how to solve it stepbystep.. can someone show me pls?
  19. R

    Antiderivatives math problem

    Find the following. ∫ (2y^(1/2) - 3y^(2)) / 6y ; dy It's number 34 if you want to see it.Thanks. http://pic20.picturetrail.com/VOL1370/5671323/23643016/396306428.jpg I did ∫ [ ( 2√y - 3y² ) / ( 6y ) ] dy = ∫ { [ (2√y ) / (6y) ] - [ (3y²) / (6y) ] } dy = (1/3) ∫ ( 1/ √y ) dy - (1/2) ∫...
  20. R

    Antiderivatives. Did I do it right?

    Find the following. ∫ (56t^(5/2) + 18t^(7/2))dt. I did (2/7)56 t^(7/2) + (2/9)18t^(9/2) + c = 16t^(7/2) + 4t^(9/2) + c
  21. R

    Is this the correct way to find the antiderivatives?

    Find the following. ∫ (v^2 - e^(3v)) dv. I did ∫(V^2-e^(3v)) dv ∫(v^2)dv - I (e^(3v) )dv ∫(v^3)/3- (e^(3v))/3 ∫(v^3-e^(3v))/3 Did I so it right?
  22. 3

    Integration and antiderivatives in class

    Ive just started learning about integration and antiderivatives in class, and I've got a question Say we have: f(x) = 1/x^3 , and f'(x) = F(x) g(x) = 1/X , and g'(x) = G(x) then to find the antiderivative of f(x), i would solve it like this: first rewrite it as : x^(-3), = x^(-3 + 1) / (...
  23. T

    Differential forms as antiderivatives?

    Hi, I had a silly idea that probably doesn't work, but I thought I'd ask about it anyway. I understand that vectors can be thought of as derivative operators, e.g. \frac{d}{d\lambda} = \frac{dx^\mu}{d\lambda} \partial_\mu, where lambda parametrizes some curve. I also gather that one-forms...
  24. M

    Antiderivatives & Indefinite Integrals

    Homework Statement Compute the following integral: Homework Equations none The Attempt at a Solution x^(1/2)-2(x^2)^(1/3) +1*x^(-1/4)dx = (2/3)x^(3/2)-...I don't know this part...+x * (4/3)x^(3/4) and this is where I stopped... btw sorry I didn't put the symbols and...
  25. 0

    Antiderivatives (Finding f(x), C and D)

    Homework Statement Find f for: f double prime (x) = 2x^3 + 3x^2 - 4x + 5, f(0) = 2, f(1) = 0 Homework Equations Most general antiderivative: F(x) + C The Attempt at a Solution = F(x) + C = f prime (x) = [(2x^4)/4] + [(3x^3) / 3] - [(4x^2)/2] + 5x + Cx = (1/2)(x^4) + x^3 - 2x^2...
  26. 0

    Antiderivatives (Is my book incorrect?)

    Is my working correct, or is my math pdf (in red font) correct? 1. The problem statement: Find f(x) given that f′′(x) = 3/squareroot of x, f(4) = 20 and f′(4) = 7. 2. Most General Antiderivative: F(X) + C 3. The Attempt at a Solution First, simplify 3/squareroot of x: =...
  27. 0

    Antiderivatives (where did I go wrong?)

    Where did I go wrong with my working? The answer in the book is f(t) = t^2 + 3 cos t + 2 1. The problem statement: Find f for f prime (t) = 2t - 3 sin t, f(0) = 5 2. Homework Equations : Most General antiderivation: F(x) + C Antidifferentiation formula: Function = sin x, Particular...
  28. 0

    Antiderivatives (find f for )

    Antiderivatives - 8x - 3 sec^2x Im not sure whether my answer to this antiderivative question is correct and would like your opinion. 1. The antiderivative statement asks: Find f 2. f prime (x) = 8x - 3 sec^2x Most general antiderivative = F(x) + C Antidefferentiation formula: Function...
  29. 0

    What is the Antiderivative of (f(x) = √(x^5 ) - 4/√(5&x))?

    1. This antiderivative question asks: Find f. The equation reads: f prime of x = square root of x to the 5th power minus 4 divided by the fifth root of x. (see below) f´ (x) = √(x^5 ) - 4/√(5&x) The answer in the back of the book is f(x) = (2/7)X^(7/2) - 5X^(4/5) + C but I got stuck...
  30. S

    Antiderivatives question

    Homework Statement 1.∫sec^2 (4x+1)dx 2.∫ root(sin pi theta) cos(pi theta) d(theta) 3.∫ e^x dx /(1+e^x) Homework Equations 1.u=4x+1 2. u= sin(pi theta) 3. u= 1+e^x antiderivatives to find the answers.. The Attempt at a Solution For #1, tan^4(4x+1)/4 +c..is the answer..but I...
  31. M

    Improper Integrals / Antiderivatives.

    Homework Statement Evaluate each improper integral whenever it is convergent. 1. S 1-infinity 4 / x 2. S 4 - infinity 2/x^3/2 Homework Equations The Attempt at a Solution I'm having trouble with antiderivatives. I understand how to do them when the problem is like x^4 + x^2...
  32. V

    Logs and antiderivatives

    S (x+2)/(x^2+4x) dx I've been learning about natural logs and their properties but at this answer I get befuddled at how to work it out. I think perhaps substitution, and some properties with logs but I am very weak in logs...Didn't do well in it when I was in algebra. I want to know how to...
  33. U

    Antiderivatives Graphically & Numerically

    Homework Statement Using the graph of g’ in the figure below and the fact that g(0) = 50, sketch the graph of g(x). Give the coordinates of all critical points and inflection points of g. http://s52.photobucket.com/albums/g14/CCieslak1689/?action=view&current=graph.jpg 2. Attempt and...
  34. W

    Solve Some Antiderivatives: A Math Homework Statement

    [SOLVED] Some antiderivatives I've got a few antiderivatives to find, I've found most of them and they check out fine with my CAS, but three of them I'm having difficulties with. The first: Homework Statement I = \int {{{\sec ^2 \left( x \right)} \over {\left( {1 + \tan \left( x \right)}...
  35. Y

    Calculating Distance Traveled Using Antiderivatives: Solving for k

    Homework Statement A car going 70km/h comes to a stop in 6 seconds, assume that the deceleration is constant, find the distance traveled using a graph; find the distance traveled using antiderivatives. The Attempt at a Solution If the deceleration is constant, I would assume that the...
  36. A

    Constructing Antiderivatives

    Homework Statement A 727 jet needs to attain a speed of 200 mph to take off. It it can accelerate from 0 to 200 mph in 30 seconds, how long must the runway be? The Attempt at a Solution First I converted mi/hr to mi/sec 200mi/hr * 1hr/60min * 1min/60sec = 0.056 mi/sec and since...
  37. A

    How long does it take for a car to travel 100 meters with a given acceleration?

    Homework Statement A car starts from rest at time t=0 and accelerates at -0.6t+4 meters/sec^2 for 0\leqt\leq12. How long does it talke for the car to go 100 meters? The Attempt at a Solution Since it says the acceleration is -0.6t+4 I integrated it and ended up with v(t)=-0.3t^2 + 4t...
  38. daniel_i_l

    Can c_1 and c_2 Make F an Antiderivative of f?

    Homework Statement Let f:[0,2]->R be defined as: if 0 =< x =< 1 then f(x) = 4(x^3) if 1 < x =< 2 then x = x^2 + 2 Prove or disprove: There exist c_1 , c_2 in R so that F:[0,2]-R defined as: if 0 =< x =< 1 then f(x) = x^4 + c_1 if 1 < x =< 2 then x = (x^3)/3 + 2x + c_2 Homework...
  39. E

    Two integrals I am trying to solve without closed form antiderivatives

    How do I solve the integral of the functions x*exp(-a(x-b)^2) and x^2(exp(-a(x-b)^2) where a and b are positive real numbers? I tried integration by parts and cannot think of how to find the integral. In addition, while I was able to find exp(-a(x-b)^2) in an integral table, the two functions...
  40. J

    Constructing Antiderivatives and areas

    Homework Statement The origin and the point (a, a) are at opposite corners of a square. Calculate the ratio of the areas of the two parts into which the curve \sqrt{x} + \sqrt{y} = \sqrt{a} divides the square.Homework Equations I'm sure there will be some use of A = bh. Perhaps maybe the...
  41. C

    Graphical Antiderivatives

    A tow truck is pulling a car. Initially the force that the truck applies to the car is quite large. However, it decreases linearly to zero and remains at zero. Describe the how the acceleration, the velocity and the position change in time during this process. Ok I have no idea where to...
  42. E

    Learn Antiderivatives of x^n: C1 & C2 Explained

    I'm learning about antiderivatives now, and I just learned a formula to find the antiderivative of xn: if f(x) = xn, and F'(x) = f(x), then F(x) = (xn+1)/(n+1) + C, where C is a constant but if n < 0, then F(x) = (xn+1)/(n+1) + C1, if x > 0 F(x) = (xn+1)/(n+1) + C2, if x < 0 My question is why...
  43. P

    Power rule for antiderivatives

    I am taking an Architectural Geometry class, and have only had Precal. We just started antiderivatives (I understand regular derivatives), and had a question: I have to find the antiderivative of (-5/12 x^4) + (10/3 x^3) - (103/12 x^2) + (23/3 x) I think I use the power rule for...
  44. M

    Finding representations of antiderivatives without a closed form

    I was wondering if anyone knew of any good books (or textbooks, or websites) which discuss finding series representations of integrals which exist, but don't have a closed form. I'm interested in the subject at the moment, but I haven't had much luck online. Furthermore, what branch of calculus...
  45. M

    Determining the existence of antiderivatives

    Ok, I realize that this question is rather broad, but how can one determine whether a function has an antiderivative or not? I know that for an arbitrary function, chances are that it won't have a "nice" one. The function e^(-x²) comes to my mind. However, if I understand it correctly, its...
  46. F

    I on the graphs of antiderivatives

    Hello everyone. I need help. If there are given 2 graphs in which one of them is the graph of the derivative of the other graph (that graph is the antiderivative - the "original" one), how can I tell the graph of a derivative from a graph of an antiderivative? In another case: the...