Antiderivative Definition and 177 Threads

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. Ackbach

    MHB What Is the Antiderivative of 1/(1 + x^4)?

    Problem: find $$\int \frac{dx}{1+x^{4}}.$$ Answer: It's perhaps a not-so-well-known fact that, although $x^{2}+y^{2}$ does not factor over the reals, $x^{4}+y^{4}$ does. In fact, $$x^{4}+y^{4}= \left(x^{2}+ \sqrt{2} \, xy+y^{2} \right) \left(x^{2}- \sqrt{2} \, xy+y^{2} \right).$$ Hence, we have...
  2. P

    The Tricky Antiderivative: Solving the Integral of a Radical Function

    1. The problem, the whole problem, and nothing but the problem $$ \int \sqrt{\frac{1+x^2}{1-x^2}} dx $$ Homework Equations ? The Attempt at a Solution I tried a partial fraction decomp, but this obviously didn't get me very far because this isn't a rational function. Is there some kind of...
  3. P

    Solving Hard Antiderivative: u-substitution & Trig Substitution

    1. The problem, the whole problem, and nothing but the problem \int \frac{dx}{ \sqrt{ 1+ \sqrt{ 1+ \sqrt{ x } } } } Homework Equations u-substitution (in the style of trig substitution) I think that I've got it figured out, I just don't know if my substitutions were legitimate...
  4. D

    Antiderivative and Indefinite Integration

    Homework Statement ∫(x3 - 3x2 + x + 1)/√x dx The Attempt at a Solution ∫x3-1/2 - 3∫x2-1/2 + ∫x1-1/2 + ∫x1-1/2 ∫x5/2 - 3∫x3/2 + ∫x1/2 + ∫1/2 (x7/2)7/2 - (3x5/2)5/2 + (x3/2)3/2 + (x3/2)3/2 + C (2x7/2)/7 - (6x5/2)/5 + 2x3/2)/3 + (2x3/2)/3 + C Thank you so much!
  5. Mandelbroth

    Derivation of the Antiderivative of the Gaussian Distribution

    I'm in a high school pre-calculus class and a statistics class. For the latter, we are given z-tables for some of our tests. I don't like these z-tables. Thus, I decided that a more direct approach (fundamental theorem of calculus) would be more accurate and, more importantly, more fun. My...
  6. PhizKid

    Finding antiderivative without integration

    Homework Statement f'(u) = 1 / (1 + u^3) g(x) = f(x^2) Find g'(x) and g'(2) Homework Equations The Attempt at a Solution So the derivative of function f at u is: 1 / (1 + u^3) That means g'(x) would be f'(x^2), but to find the general derivative of f at u is 1 / (1 + u^3) so can...
  7. I

    G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)

    G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)... Find a formula for G(x) in terms of F(x). I would have just written \int_0^x{F(x)dx} but I know the value G(1) = 1 has to be included somewhere. Can someone help me figure this one out?
  8. D

    Find a formula for a higher degree antiderivative

    The problem asks to find a formula for a higher degree antiderivative. This formula pattern is similar to the one stated in the Fundamental Theorem of Calculus: F(X)=∫f(t)dt. Fn(x)=∫*F(t)dt, with certain expression in the asterisk.
  9. B

    What Is the Correct Antiderivative of 1/(x^2)?

    Homework Statement find the antiderivative of 1/(x^2) The Attempt at a Solution I'm pretty sure you just find the antiderivatives of the numerator and the denominator. the antiderivative of 1 is x. the antiderivative of x^2 is (x^3)/3 mutliply the numerator by the inverse of...
  10. P

    Antiderivative Help: Solving ∫(1+sin(x))/(cos^2(x)) dx

    Homework Statement Here is the problem: ∫(1+sin(x))/(cos^2(x)) dx Also- how do you guys type equations in this? The quick symbols doesn't have a fraction bar or definite integral... Homework Equations sin^2(x)+cos^2(x)=1 The Attempt at a Solution I substituted 1-sin^2(x) for cos^2(x) and...
  11. F

    What is the antiderivative of x^x?

    The derivative of x^x is x^x(lnx+1) but what would be its antiderivative? I don't think the answer is in elementary terms. According to someone there is a special function made just to answer this question. So what is it?
  12. P

    How is U(x) integrable if it doesn't have an antiderivative?

    Homework Statement Show that the function U(x) defined by U(x)=\begin{cases} 0 & \text{ if } x<0 \\ 1 & \text{ if } x\geq0 \end{cases} has not an antiderivative in (-∞, +∞). (Suggestion: Assume U has an antiderivative F in (-∞, +∞) and obtain a contradiction, showing that this...
  13. X

    Integral Calculus Antiderivative Question

    Homework Statement Find the antiderivative of: \int \frac{dx}{x^3 - 2x^2 + 4x - 18} Homework Equations This was asked in my Calculus II class right after we finished dealing with Solving for Integrals using Partial Fractioning. The Attempt at a Solution This is really more of an algebra...
  14. H

    Integration with x denominator plus a few antiderivative concept questions

    Homework Statement v(t) = (-2 / (1 + 2t)) + 2 I must find the antiderivative s(t) such that s(0) = 5. The Attempt at a Solution u = 1 + 2t du/dt = 2 dt = du/2 √(-2 / (1 + 2t dt)) + 2 =√(-2u^-1 du/2) + 2 =√(-u^-1 du) + 2 = -ln|1+2t| + 2t + C At t = 0 -ln(1 +...
  15. T

    Homework: Finding an antiderivative

    Homework Statement dy/dx = y(1-y) Homework Equations The Attempt at a Solution dx = dy(y(1-y))?
  16. F

    Antiderivative of an even function

    Is the antiderivative of an even function even? Specifically, given an even function g(x)=g(-x) will the following function be even? Odd? For constant a>0, r a dummy variable h(x,t) = Definite Integral [ g(r) dr, FROM x-at TO x+at] Asking whether or not h(x,t)=h(-x,t) This is the...
  17. N

    How Do You Calculate the Area Under a Curve Using Antiderivatives?

    Homework Statement As part of a bigger problem, I am trying to find the area under the curve f(x) = sqrt{2x - x2} between x = 0 and x = 2. To do this, I have to find the antiderivative of f(x) Homework Equations antiderivative f(x) = axb F(x) = a/(b+1) * xb+1 chain rule f(x) = a(b(x)) f'(x)...
  18. K

    What is the Antiderivative of Inverse Sine of x?

    Homework Statement Find the antiderivative of \frac{1}{\sin\alpha} Homework Equations The Attempt at a Solution Ugh, maybe it's just a temporary brainfreeze, but I have no idea how to go about it.
  19. R

    Complex analysis antiderivative existence

    Homework Statement a) Does f(z)=1/z have an antiderivative over C/(0,0)? b) Does f(z)=(1/z)^n have an antiderivative over C/(0,0), n integer and not equal to 1. Homework Equations The Attempt at a Solution a) No. Integrating over C= the unit circle gives us 2*pi*i. So for at least one...
  20. P

    Why 1/z dont have antiderivative?

    Homework Statement why \frac{1}{z} don't have derivative? i know that ( \log z )'=\frac{1}{z} so why \int \frac{1}{z} \mbox{d} z don't exist? Homework Equations The Attempt at a Solution
  21. P

    Does an antiderivative of e^z/z^3 exist in the punctured complex plane?

    Hi, so my question is the subject line. In the multiply connected domain |z|>0, does the function f(z) = e^z/z^3 have an antiderivative? I'm learning from Brown and Churchill, and they have a theroem on pg. 142 that leads me to believe it does. I don't remember what my prof said about this...
  22. R

    What is the Antiderivative of a Complex Square Root Function?

    Homework Statement Antiderivative of sqrt[e^x+ln(x^2)+1] Homework Equations The Attempt at a Solution (2[(e^x+ln(x^2)+1)^(3/2)]x^2)/[3(e^x+2x)]
  23. S

    Antiderivative of 1/((x(lnx)^3) using u-substitution

    Homework Statement I'm working on an infinite series problem and need to find the antiderivative of 1/((x(lnx)^3). Homework Equations u=lnx The Attempt at a Solution I know I have to use the substitution u=lnx, but I still can't figure out what the answer is. I know the...
  24. B

    Quick Calc 1 problem (Antiderivative)

    Hello, I'm looking for f given f'' and two conditions. [PLAIN]http://img375.imageshack.us/img375/2572/antider.jpg Going from f'' to f', I get f'(x) = 5x^4 + 4x^3 + 4x + C But with the two conditions, I feel that I cannot progress from here. I feel that one of those conditions should be...
  25. C

    How to calculate the antiderivative of: [sqrt(t^4 + t^2 + 1)]/(t^2 + 1)

    Anybodys knows how to resolve the problem of the subject. I am working in it for three hours. I am desperated. I have tried all the subtitutions of "Calculus. Schaum. Ayres&Mendelson". Whithout suceeds. Thank you. Diego.
  26. E

    Antiderivative of 6e^x and 4x^2

    antiderivative of 6e^x and 4x^2 --> 6e^x -->1/6e^x?? --> 4x^2 --> 2x^3??
  27. W

    What is the antiderivative of cosh(x^2)?

    Homework Statement Just trying to figure out the anti-derivative of cosh(x^2). Homework Equations I knowthe antiderivative cannot be expressed as an elementary function but I am pretty clueless of getting the antiderivative though! The Attempt at a Solution I am baffled by this one...
  28. V

    Antiderivative homework question

    Basically I need to know \int_0^h x\frac{1}{2\sqrt{hx}}dx my working, \frac{1}{2\sqrt{hx}} \int_0^h \frac{x}{\sqrt{x}} , how do i do this?, \int \frac{1}{\sqrt{x}} \int x^(\frac{-1}{2}) \frac{1}{\frac{-1}{2}+1} x^(\frac{-1}{2}+1) \frac{x^\frac{1/2}}{\frac{1}{2}} 2x^\frac{1}{2}...
  29. N

    Antiderivative math homework help

    Homework Statement \int(2x^2+1)^7 Homework Equations The Attempt at a Solution u=2x^2+1 du=4xdx u7 (1/4x)du I am stuck... I don't know what to do next...
  30. K

    Area from antiderivative of curve to x-axis

    I'm trying to find the area between the antiderivative of the curve y = 5x3 - 23x2 - x + 3 passing through the point (1, e) and the x-axis from x = 1/√2 to x = 17π/11, to 3 significant figures. I'm self teaching myself calculus, and this has me stumped.
  31. L

    Can x^2/2 + y Be Considered an Antiderivative of x?

    x^2/2 is an antiderivative of x, for the derivative of x^2/2 with respect to x is x. Formally speaking, can I consider x^2/2 + y, where y is a variable and not a constant, to be an antiderivative of x, since the partial derivative of x^2/2 + y with respect to x equals x?
  32. mnb96

    Antiderivative of multivariable function?

    Hello, if I have the following unknown function f(x_1,\ldots,x_n) Assuming I am given all its partial derivatives \frac{\partial f}{\partial x_i} is it possible to get the original function f ? This is clearly possible for a one-variable function f(x). If we know df/dx we just need to...
  33. S

    Antiderivative graphing question - help

    Homework Statement http://img64.imageshack.us/img64/5430/80433637.jpg Homework Equations The Attempt at a Solution for a, would it be local.max: x=7.7 and local.min: x=4.8 ? and for b, abs.max: x=11 and abs.min: x=2 ? im not sure if i did it right but looks like this isn't...
  34. S

    Antiderivative of a function problem

    Homework Statement $g(x)=\int _{2 }^{\sin x}\sqrt{1- t^2}dt$ whats g'(x)...Homework Equations The Attempt at a Solution how to find the antiderivative of sqrt(1-t^2)?
  35. O

    Finding the antiderivative (inverse) of a function

    Homework Statement f(x)=1+e^x/1-e^xHomework Equations truth is, I don't even know how to approach this, i know i have to swap the variables but now I'm all confused because a friend told me to do this, in this particular case f(x)=[1+(e^f)-1^x]/[1-(e^f)-1^x] The Attempt at a Solution i don't...
  36. K

    Differential Galois Theory: exp(-x^2) has no elementary antiderivative

    A lot of apparently innocent elementary functions, like exp(-x^2) or (sin x)/x, have not antiderivatives in terms of elementary functions. I've read that "Differential Galois theory" explains this, and gives an algorithmic method to know if a given elementary function has or has not elementary...
  37. G

    Is a most general antiderivative required to be continous?

    I've looked through my textbook and I've found no mention, but I've seen some posts on the Internet which explicitly state to check for continuity while taking the derivative certain functions. Is it because: F'(x) = f(x) I understand that a function is required to be continuous and...
  38. A

    A difficult integration (antiderivative) problem

    Homework Statement Decide whether f(x)=\int (1-cos(x))/x^2 is improperly integrable on (0, infinity). Homework Equations The Attempt at a Solution I understand the concept of improper integration, but I don't see how to take the antiderivative -- I tried substitution and by parts...
  39. G

    Antiderivative as Area under Curve

    The proof I'm familiar with for relating the antiderivative to the area under a curve involves usage of the mean value theorem, which for that particular case, implies continuity for the curve. Thus, integration as a process for finding the area under a curve should be valid under the...
  40. C

    How Does the Chain Rule Apply to Finding Antiderivatives in the Volume Equation?

    The text is giving me steps on how to do it but I don't get what's it is asking. The equation is (v^-.5)v'=k where v= volume Then says we consider the chain rule together with elementary antiderivative formulas to determine an antiderivative with respect to t of (v^-.5)v' I haven't...
  41. E

    Quick Antiderivative of Tan^2x and Sec^2x

    Homework Statement Find the antiderivative of y=tan^2x+sec^2x Homework Equations N/A The Attempt at a Solution Seems to be a simple question, but the answer is eluding me no matter what I do. My first try was to replace the tan^2 with sec^2-1, and then factor out a 1/2 from the...
  42. J

    Having trouble with the antiderivative of this function

    Homework Statement Evaluate the indefinite integral: \int7x+1/x^2+1 dxHomework Equations The Attempt at a Solution My first attempt at the solution was to try using substitution. I set u=x^2+1. so du=2x dx and x=sqrt(u-1). Then I rewrote the integral so it is \int7du/4usqrt(u-1). This is...
  43. A

    How Do You Integrate Trigonometric Vectors with Variable Substitution?

    3 sin ^2 t cos t (i) + 3 sin t cos^2 t (j) + 2 sin t cost (k) I have to take the antiderivate for each Vector. Then I have to evaluate it from pi/2 to o. I'm confused because I can't use a trigonometric substitution. Cosine is odd for the I vector but I can't substitute 1-sin^2...
  44. V

    Antiderivative of anti-trigs(arctan)

    Homework Statement arctan(x)/(1+x2). finding the antiderivative Homework Equations arctanx=1/1+x^2 The Attempt at a Solution I tried pulling out arctan(x) S 1/(1+x^2) dx --> arctan^2(x)+c but the answer also has a 1/2 in front of the arctan...where'd that come from?
  45. M

    What is the antiderivative of e^ln(4)x?

    Homework Statement what is the antiderivative of e^ln(4)x Homework Equations The Attempt at a Solution Ok someone told me that the antiderivative of 4^x is e^ln(4)x. So what would be the antiderivative of e^ln(4)x? I need these antiderivatives in order to solve my homwork...
  46. C

    Solve Antiderivative Problem - Step-by-Step Guide

    edit: nvm. i'll figure it out... :/
  47. L

    Need help finding an antiderivative

    need help finding an antiderivative! Homework Statement 1 ∫ 36/ (2x + 1) ^3 dx 0 I just can't figure out how to take the antiderivative of this! What do I do with the (2x +1) ^3?? Homework Equations The Attempt at a Solution
  48. C

    Beginning Trignometric Antiderivative

    Homework Statement Okay, I think I'm finally getting the hang of these antiderivatives. However, I'm still stumbling some on trigonometric functions. Find the antiderivative of f(\theta) \ = \ \frac{1 + \cos^2{\theta}}{\cos^2{\theta}} Homework Equations f(\theta) \ = \ \frac{1 +...
  49. C

    Beginning Antiderivative Question

    Homework Statement Find the antiderivative of F(x) \ = \ \sqrt{\frac{5}{x}} Homework Equations F(x) \ = \ \sqrt{\frac{5}{x}} The Attempt at a Solution 1.) F(x) \ = \ \sqrt{\frac{5}{x}} 2.) F(x) \ = \ (\frac{5}{x})^\frac{1}{2} 3.) F(x) \ = \ \frac{5^\frac{1}{2}}{x^\frac{1}{2}}...
  50. M

    Finding the antiderivative of a fractional prob

    I am suppose to evaluate this integral by using substitution f'(x)=(e^-0.5x)/(1-e^-.5x) I will be very thankful if someone were able to tell me a strategy in antiderivitating fractional problems. What I did: I set u=-0.5x dx=du/-0.5 therefore i got (-1/0.5)((e^u)/(1-e^u))du I...
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