What is Antiderivative: Definition and 177 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. S

    I Did Math DF's integral calculator make a glaring mistake?

    In calculating the integral ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}##, I used a few online integral calculators to check my answer. According to one calculator, I got the correct antiderivative, but according to another (Math DF Integral Calculator)...
  2. S

    Antiderivative of Heaviside step function with absolute-value-argument

    For ##R<0##, the antiderivative is just a constant, since then ##R-|x|## is negative for all values of ##x##, which in turn implies ##\Theta(R-|x|)## is zero for all values of ##x##. For ##R\geq 0##, and by inspection apparently, the antiderivative is ##(R+x)\Theta(R-|x|)+2R\Theta(x-R)+C.##...
  3. R

    Why don't we use the antiderivative factor for Average Voltage

    Below is the really quick derivation for average voltage. However when we do the anti-derivative a factor of ##1/\omega## should come out and the full answer should be ##\frac{2V_p}{\pi \omega}##. So why don't we include that? What's going on?
  4. SchroedingersLion

    From the limit of the derivative, infer the behavior of the antiderivative

    Greetings! In statistical mechanics, when studying diffusion processes, one often finds the following reasoning: Suppose there is a strictly positive differentiable function ##f: \mathbb{R} \rightarrow \mathbb{R}## with ## \lim_{x \rightarrow +\infty} {f'(x)} = a > 0##. Then for sufficiently...
  5. Abdullah Almosalami

    I Is there such a thing as an antiderivative of a multivariable function?

    Is there such a thing as an antiderivative of a multivariable function? I haven't put too much thought into this yet but I wanted to ask anyways. Sticking for now just to two variables, I was observing that double integrals are always definite integrals, whereas in the single-variable case, we...
  6. lfdahl

    MHB Find the antiderivative of V(2−x−x^2)/x^2

    Find the antiderivative, $F$, of the function $$f(x) = \frac{\sqrt{2-x-x^2}}{x^2}.$$
  7. 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...
  8. P

    MHB Berk's question via email about an antiderivative

    This requires using Integration By Parts twice... $\displaystyle \begin{align*} I &= \int{\mathrm{e}^{-2\,x}\cos{ \left( 3\,x \right) } \,\mathrm{d}x} \\ I &= \frac{1}{3}\,\mathrm{e}^{-2\,x} \sin{(3\,x)} - \int{ -\frac{2}{3}\,\mathrm{e}^{-2\,x} \sin{(3\,x)}\,\mathrm{d}x } \\ I &=...
  9. M

    Conditions on f such that the anti-derivative F is periodic?

    Homework Statement Let f : R → Rn be a smooth function. Give necessary and sufficient conditions on f so that the antiderivative F(x) = ∫f(t)dt (from 0 to x) is periodic with period p ≠ 0 Homework EquationsThe Attempt at a Solution My initial thought is that as long as f is periodic then F...
  10. lfdahl

    MHB What is the antiderivative of $\frac{1}{x^{1/2}+x^{1/3}}$ for $x>0$?

    Find the antiderivative ($ x > 0$):\[\int\frac{1}{x^{1/2}+x^{1/3}}dx\]
  11. Alettix

    Antiderivative of 1/x: ln(x) or ln(|x|)?

    Homework Statement Calculate the integral: ## \int_{a}^{b} \frac{1}{x} dx ## Homework Equations - The Attempt at a Solution In high school we learned that: ## \int_{a}^{b} \frac{1}{x} dx = ln(|x|) + C ## because the logarithm of a negative number is undefined. However, in my current maths...
  12. Rectifier

    Finding anitderivative using complex numbers and Euler

    I have to find a primitive function below using the Euler formulas for ##\sin x## and ## \cos x## The problem $$ \int e^{2x} \sin 3x \ dx $$ Relevant equations ## \cos x = \frac{e^{ix}+e^{-ix}}{2} \\ \sin x = \frac{e^{ix}-e^{-ix}}{2i} \\ \\ \int e^{ix} \ dx = \frac{e^{ix}}{i} ## The attempt...
  13. Rectifier

    Antiderivative of a rational function

    I am trying to find primitives to the rational function below but my answer differs from the answer in the book only slightly and now, I am asking for your help to find the error in my solution. This solution is long since I try to include all the steps in the process. The problem $$ \int...
  14. 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
  15. U

    MHB Find the derivative of the antiderivative

    I know it seems pretty self explanatory, but I've tried to do this question and I've apparently gotten the wrong answer twice. If anyone can give me a clear solution to the problem, that would be greatly aooreciated. I initially tried to follow a video I saw online, but I think there is...
  16. WeiShan Ng

    Find Antiderivative of y: y^2=x^2+1

    Homework Statement x=sec(t), y=tan(t), -π/2 ≤ t ≤ π/2 Try to find y in terms of x Homework EquationsThe Attempt at a Solution 1.[/B] ∂y/∂x = sec(t)/tan(t) y=∫sec(t)/tan(t)∂x =∫x/y∂x =(1/y)*∫x∂x =x2/2y + C 2y2=x2 + C When t=π/4, x=√2, y=1 2(1)2 = (√2)2 + C C=0 So y2 = x2/2 2. y/x = sin(t)...
  17. egio

    I Difference between integral and antiderivative?

    Hello, I still don't really understand what an antiderivative is, besides its ability to "undo" derivatives, its relation to integrals, and what the difference between the two even is. It would also be great to know how to visualize an antiderivative. I've tried looking further into the...
  18. I

    Antiderivative and contour integration

    I'm wondering if this could be used to calculate the value of a contour integral directly. If a function has an antiderivative on the entire complex plane, this implies the field is conservative so we should always get the same value no matter which path taken. Shouldn't this also mean integrals...
  19. C

    Find Antiderivative Help: ∫[(3x+1)/cos^2(3x^2+2x)]dx

    Homework Statement Find the antiderivative: ∫[(3x+1)/cos^2(3x^2+2x)]dx Homework EquationsThe Attempt at a Solution I attempted to use "u substitution" but got stuck towards the end: u=3x^2+2x du=6x+2dx =2(3x+1)dx du/2=(3x+1)dx After my substitutions it looks like this: ∫(du)/2cos^2(u) =...
  20. O

    Finding Complex Antiderivatives | Guidance for Tricky Functions

    Homework Statement How would one go about finding the antiderivative to this function? Homework Equations N/A The Attempt at a Solution This problem has been rather tricky I have tried several attempts at the solution. My one solution consists of me factoring out the x^4. Looking for some...
  21. J

    How do I set this antiderivative up?

    Mod note: Moved from technical math section, so there's no template. Sorry if I'm formatting this question wrong, new user. F(x) is an antiderivative of https://upload.wikimedia.org/math/9/1/5/915ca58b070b0328cd069524c2d487f2.pngx3+x+1...
  22. O

    Antiderivative Help: Calculate Integral

    Homework Statement Calculate the following integral: Homework Equations N/A The Attempt at a Solution from this point I tried a u-substitution by letting u = -3 + 4/x -1/x^2 but this seemed to fail. Are any suggestions possible?
  23. PhysicsBoyMan

    Solving Integrals with Exponents: A Step-by-Step Guide

    Homework Statement Evaluate the integral to find the area. Homework Equations The Attempt at a Solution[/B] gifs upload So I know how to find an anti-derivative for the most part. Here it's tricky because my equation has an exponent AKA square root. I tried to use the chain rule with...
  24. O

    Simplifying Trigonometric Antiderivative using Right Triangles?

    Mod note: Moved from a homework section 1. Homework Statement Hello my question more has to do with theory that perhaps deals with algebra. Why is the following true? Homework Equations N/A The Attempt at a Solution N/A[/B]
  25. Cosmophile

    What is the Antiderivative of 1/x in Calculus?

    We are going over antiderivatives in my calculus course and reached a question regarding ##f(x) = \frac {1}{x}##. My instructor went on to say that ##\int \frac {1}{x}dx = \ln |x| + C##. This makes sense to me, but only to a certain point. For ##f(x) = \frac {1}{x}##, ##f## is defined...
  26. B

    Find the antiderivative of the following vector:

    Homework Statement Calculate the position vector of a particle moving with velocity given by: v = (32 m/s - (5 m/s^2 )t i) + (0 j) Homework Equations (x^(n+1) / (n+1) ) + C = antiderivative of function The Attempt at a Solution r = (32t m - (5/2)t^2 m/s + C m i) + (C j) Honestly, I'm just...
  27. O

    How can I find the antiderivative of this complicated Bessel function?

    I am struggling to find the antiderivative of the following function: f(x)=\frac{J_{0}(ax)J_{1}(bx) }{x+x^{4} } \\ J_{0},{~}J_{1} : Bessel{~}functions{~}of{~}the{~}first{~}kind\\ a, b: constants \\ F(x)=\int_{}^{} \! f(x) \, dx =? Who can help?
  28. P

    Is the Domain of an Antiderivative Always a Subset?

    Isn't the domain of the derivative of a function a subset of the domain of the function itself? Does this mean that the domain of an integrand is always a subset of the corresponding indefinite integral?
  29. P

    MHB How Can I Find the Integral of Arcsec(x)dx Using IBP?

    Can someone help me with finding the integral of arcsec (x) dx? Thanks Prakash
  30. T

    Antiderivative of (e^sin(t)) *(cos(t))?

    Homework Statement The antiderivative of (e^sin(t)) *(cos(t)) is e^(sin(t)) + C? Why is this? What happened to the cos(t)? Is there the chain rule or something applied? I don't know! It just looked like it disappeared.
  31. 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...
  32. 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...
  33. nuuskur

    Antiderivative of trig function

    Homework Statement \int\frac{{\rm{d}}x}{1+2\sin^2 (x)} Homework EquationsThe Attempt at a Solution it's some sort of a derivative of arctan, however, when I try to substitute y = \sqrt{2}\sin(x),\ {\rm{d}}y = \sqrt{2}\cos(x){\rm{d}}x I get nowhere with it, atleast I think, since there is a...
  34. A

    Calculating Time and Distance for a Tram with Sinusoidal Velocity

    Homework Statement The velocity of a tram traveling on a straight line between two stops is given by: v = 16sin(pi(t)/30) m/s find the time lapse between stops the distance traveled between stops the maximum velocity of the tram and when it occurs Homework Equations Displacement = x...
  35. nuuskur

    Antiderivative involving a radical

    Homework Statement \int\frac{1}{(x+1)^2}\sqrt{\frac{x}{1-x}}{\rm{d}}x Homework EquationsThe Attempt at a Solution Utterly perplexed. Have no ideas how to do this one. Did try bringing the entire thing under square root and try partial fractions, but the entire thing is modified by the square...
  36. nuuskur

    Antiderivative of a rational function

    Homework Statement Consider the integral: \int\frac{2x^3 -4x^2 +8x +7}{(x-1)^2 (x^2 +4x +8)}{\rm{d}}x Homework EquationsThe Attempt at a Solution The degree of the denominator is 4 and the numerator's is 3, hence I thought I would try partial fractions: \frac{A}{x-1} +\frac{B}{(x-1)^2}...
  37. W

    Proof showing that if F is an antiderivative of f, then f must be continuous.

    Homework Statement Show that if F is an antiderivative of f on [a,b] and c is in (a,b), then f cannot have a jump or removable discontinuity at c. Hint: assume that it does and show that either F'(c) does not exist or F'(c) does not equal f(c). 2. The attempt at a solution I attempted a proof...
  38. K

    Integral, why antiderivative is area under curve

    http://www.askamathematician.com/2011/04/q-why-is-the-integralantiderivative-the-area-under-a-function/ this website says f^\prime (c) (B-A) = f(B)-f(A) or f(c) (B-A) = F(B)-F(A) (since F’ =f). but seems like this is wrong. because B-A= Δ x, f'(c)*Δ x= Δy and the area might...
  39. J

    Explicit definition for antiderivative?

    Accidentally I wrote in the wolfram f(x) = f(1/x) the the wolfram give me the solution for this equation (f(x) = Abs(log(x))). Hummmm, nice! Thus I thought: given the definition of derivative, ##f'(x) = \frac{f(x+dx)-f(x)}{dx}##, is possible to isolate f(x) in this equation? If yes, how? I...
  40. A

    Antiderivative of ((tanx)^2)((secx)^3)dx

    My teacher said to turn all tan and sec to cos and sin. I still do not understand how I can solve this. Can you give me a hint? I know what he means by turning to cos and sin, but what do I do next?
  41. S

    MHB Antiderivative involving Trig Identities.

    A little confused on something. Suppose I have the integral 2 \int 4 \sin^2x \, dx So I understand that \sin^2x = \frac{1 - \cos2x}{2} BUT we have a 4 in front of it, so shouldn't we pull the 4 out in front of the integral to get: 8 \int \frac{1 - \cos 2x}{2} \, dx then pull out the...
  42. A

    Antiderivative vs integral

    I am just stuck on the fact that the antiderivative of cos(x) is -sin(x) but the integral of cos(x) is a positive vale of sin(x). Can someone please explain this to me. I am two weeks behind on my calculus course now because I got stuck on integration by parts and don't understand anything after...
  43. J

    Primitive, antiderivative and integral

    Prays the CFT that all function f(x) can be expressed how the integral of its derivative more an initial constant:f(x)=\int_{x_0}^{x}f'(u)du+f(x_0) So, is correct affirm that integral, primitive and antiderivative are concepts differents? ie: f(x) = primitive ##\int_{x_0}^{x}f'(u)du## =...
  44. J

    Discrete antiderivative of x^n

    Homework Statement \sum_{x} x^k for k ∈ Q Homework Equations \sum_{x} x=\frac{1}{2}x^2-\frac{1}{2}x\sum_{x} x^2=\frac{1}{3}x^3-\frac{1}{2}x^2+\frac{1}{6}x\sum_{x} x^{-1}=\Psi (x) The Attempt at a Solution I don't know. There isn't way to compute the antiderivative of any function and I...
  45. C

    Antiderivative Definite Integrals

    Homework Statement So I did an entire antiderivative, and ended with this part: sec(x)tan(x) + ln|sec(x) + tan(x)| + C I have to do this with the lower bound of -pi/3 and 0. When I do it, I should be getting 2√3 + ln(2+√3) But, I'm getting (0+0)-(2*-√3 + ln(2-√3)) Which would...
  46. Q

    What are the three extrema?

    Homework Statement https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1388504_10201044108366607_730785214_n.jpg?oh=9e67700cd15429886ee87ce2eed63328&oe=528397C9 Homework Equations F(x) = ∫f(x). We can apply the second derivative test. F''(x) = f'(x) The Attempt at a Solution F''(x) is...
  47. A

    Does a discontinuous function for have an antiderivative?

    Can F'(x) =f(x) even if f is not continuous I tried making a function let f(x) =5 if x<=5 f(x)=4 if x>5 f is not continuous at 5 Then F(x) =5x x<=5 F(x) =4x+5 x>5 Clearly F is continuous at 5 but F is not differentiable at 5.. So is there a discontinuous function that has...
  48. B

    Evaluate the antiderivative as a Taylor Series

    Homework Statement Evaluate the anti derivative ∫e^x^2 dx as a Taylor Series Homework Equations \frac{f^(n)(a)}{n!}(x-a)^n The Attempt at a Solution Where do I start, I am not sure I understand the question
  49. Petrus

    MHB How do I antiderivate \frac{8x^2}{x^2+2}?

    Hello MHB, I got problem to antiderivate \frac{8x^2}{x^2+2} Regards,