What is Indefinite integral: Definition and 207 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. T

    Solve ∫(tanxsec^{2}x)dx Integral with Substitutions

    ∫(tanxsec^{2}x)dx <---Original function to integrate. I do: ∫(tanxsec^{2}x)dx ---> ∫(tanx\frac{1}{cos^{2}x})dx ---> ∫((\frac{sinx}{cosx})(\frac{1}{cos^{2}x}))dx ---> ∫(\frac{sinx}{cos^{3}x})dx ---> substitution: [t=cosx , dt=-sinxdx -> [itex]\frac{dt}{-sinx}[/itex]=dx] --->...
  2. J

    MHB Indefinite Integral $e^x/(1+x)$

    $\displaystyle \int \frac{e^x}{1+x}dx$
  3. A

    Indefinite integral involving arctan and ln

    Homework Statement ∫ 1/x arctan (lnx) dx Homework Equations The Attempt at a Solution 1.U substitution. SO u = ln x, du= 1/x dx ∫ arctan u du 2.by parts: u = arctan u du = 1/ 1+u^2 v = 1 dv = du 3. uv - ∫vdu = artan u - ∫ 1/ 1+u^2 =...
  4. J

    MHB Indefinite Integral: $\int \frac{1}{\sqrt{\sin 2x}}dx$

    $\displaystyle \int \frac{1}{\sqrt{\sin 2x}}dx$
  5. J

    Find the following indefinite integral

    Homework Statement find the indefinite integral of ∫x√(1-x^2) dx Homework Equations The Attempt at a Solution ∫x√(1-x^2) dx let x = sinθ dx = cosθ dθ now sin^2θ + cos^2θ = 1 => cosθ = √1-sin^2θ ( for the form √(1-x^2)) ∫x√(1-x^2) dx => ∫sinθ (√1-sin^2θ) cosθ dθ =>...
  6. C

    Indefinite Integral Question; e^x

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  7. C

    Substitution Method on indefinite integral

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  8. C

    Finding the indefinite integral

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  9. A

    Calculating indefinite integral

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  10. J

    Solv Indefinite Integral: x^3/sqrt[1 + x^2] dx

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  11. E

    What is the Indefinite Integral of x*arsinh(x^2)?

    Hi guys, I don't even know where to begin with this question. Find the following indefinite integral: \int x.arsinh (x^2) dx Thanks very much for any help, its much appreciated.
  12. D

    Need Help with Hard Indefinite Integral

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  13. C

    Indefinite Integral for Torque

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  14. O

    Difficult Indefinite Integral (substitution problem?)

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  15. O

    Riemann Sum from Indefinite Integral

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  16. S

    Evaluating an indefinite integral

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  17. C

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  18. A

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  19. S

    How to evaluate this indefinite integral?

    Homework Statement \int \frac{\sqrt{x^4-1}}{x^3}dx Homework Equations The Attempt at a Solution tried substituting x^4 = \sec^2 \theta to get rid of square root but it was of no use because, I got another complex integral \int \frac{\sin^2\theta}{\cos^{\frac{3}{4}}...
  20. romsofia

    Indefinite integral of sin(x)/ln(x)?

    Anyone how to integrate this? Integration by parts doesn't get me anywhere, and there isn't a maclaurin series for ln(x). Thanks! (This isn't a homework question, I just was randomly doing integrals)
  21. P

    Integral of (5x+7)^28: Step-by-Step Solution

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  22. Z

    Finding the Indefinite Integral of a Radical Expression

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  23. M

    Is this indefinite integral really unsolvable?

    Hey, all. Anyway, browsing the Internet a bit I found this integral: \int \sqrt{1 + \frac{\ln x}{x}}dx as a proposed problem in a compilation of maths problems, as an integral from the MATYC journal. I gave it to Mathematica and WolframAlpha and they weren't able to solve it...
  24. M

    What _is_ an indefinite integral?

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  25. W

    Stuck in Evaluating an Indefinite Integral

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  26. C

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  27. M

    How Do You Integrate e^-x / (1 + e^-x) dx?

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  28. S

    Solve Indefinite Integral: 1/(x(sqrt(x^2 - 4)))

    Homework Statement integral 1/(x(sqrt(x^2 - 4))) Homework Equations I don't know if there are any "equations" for integrals... The Attempt at a Solution Int(1/(x(Sqrt(4(x^2 /4)-1) Int(1/(2x(Sqrt((x^2 /4)-1) 1/2 int(1/(x(Sqrt((x /2)^2)-1) U-sub u=x/2 du=1/2 dx 2du= dx...
  29. R

    Indefinite Integral of a contant function

    This question is bugging me if an indefinite integral is of the form F(x) + C stating that F(x) is an antiderivative of f(x) and since the derivative of a constant is 0 the collection of all of the anti derivatives are of the F(x) + C accounting for the fact any constant can be tagged to the...
  30. S

    Indefinite integral for the square root of a quartic function

    I arrived at this problem while trying to find the length of a 3D http://en.wikipedia.org/wiki/Cubic_interpolation" . Basically, I'm having to figure out how to integrate ((At4+Bt3+Ct2+Dt+E)1/2)dt A,B,C,D, and E are constants which have to be plugged in after integration, I'm sorry to say...
  31. M

    Mastering Indefinite Integrals - Essential Techniques and Examples

  32. Z

    Indefinite Integral Homework: Get a Hint to Evaluate It

    Homework Statement How do I evaluate this? A hint will do.
  33. S

    Express the indefinite integral as a power series.

    Hello, I'm kind of stuck in this problem. I have to express the integral as a power series. the integral of (e^x -1)/x I thought about evaluating it as f(x)=(e^x -1)/x and treating it as a Taylor series is that correct? Could I have any other hints? I would really appreciate it...
  34. P

    Substitution in Indefinite Integral

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  35. P

    Find the general indefinite integral

    Homework Statement Hi, can you please check the two problems? (I have never done such before) Homework Equations The Attempt at a Solution
  36. P

    Find the general indefinite integral.

    Homework Statement Hi! Can anyone check if I got the right answer? The Attempt at a Solution
  37. N

    Another Indefinite Integral question

    Homework Statement ∫cos(x)/sin^2(x)*dx Homework Equations The Attempt at a Solution Based off my earlier question, where is my error please. u=sinx du=cosx*dx ∫u^-1*du sin(x)^-1 1/sin(x) + C -or-csc(x) + C Thanks
  38. N

    Indefinite integral homework problem

    Homework Statement ∫cos(x)sin^6(x)*dx Homework Equations The Attempt at a Solution u=sin(x) then du should be -cos(x) according to the integral tables? -∫u^6*du = -u^7/7 -sin^7(x)/7+c yet my book and calculator tell me the answer is positive. How?
  39. C

    Help finding the indefinite integral

    1/(4+3x^2) ? I'm trying to use this integration formula to solve for this du ---------- a^2+u^2 I know my a would be 2, but what would my u be? I tried u = 3x and du=3dx and du/3=dx but (3x)^2 would be 9x^2
  40. N

    How can I evaluate the indefinite integral of dx/x^2*sqrt(4-x^2)?

    Homework Statement The problem reads(from the 4th edition of Stewart Calculus "Concepts and Contexts" pg. 394 #17): Evaluate the integral. the indefinite integral of dx/x^2*sqrt(4-x^2) so this would read out as "the indefinite integral of dx over x squared times the square root of 4 minus...
  41. P

    Simple Indefinite Integral of \int\frac{4}{-e^{4x-7}} = ln(-e^{4x-7}) = -4x+7+C

    \int\frac{4}{-e^{4x-7}} =ln-e^{4x-7}} =-4x+7+C The answer says it is: =\-e^{-4x+7}+C
  42. S

    Integrate S x^2(e^((x^3)+1)) with Parts - Step-by-Step

    S x^2(e^((x^3)+1)) I know I have to use integration by parts, and I'm guessing I should find the derivative of e^((x^3)+1), but I really have no idea where to start...
  43. F

    Long division to solve indefinite integral

    Homework Statement how to solve the integral of x^3 - 3x^2 + 4x -9 / x^2 + 3 dx Homework Equations using long division The Attempt at a Solution x^2 + 3 divided into x^3 - 3x^2 + 4x -9 I am not sure
  44. F

    Indefinite Integral of 1/x^2/3(1+x^1/3) | Homework Equations & Solution

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  45. B

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  46. P

    Indefinite integral (Hermite polynomials)

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  47. C

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  48. W

    Indefinite Integral Calculus II

    Homework Statement \int \frac{ \tan x \sec^2 x }{ \tan^2 x + 6 \tan x + 8 } dx Homework Equations The Attempt at a Solution \int \frac{ \tan x \sec^2 x }{ \tan^2 x + 6 \tan x + 8 } dx Okay I let... u=tanx du=sec^2x Then I got \int \frac{ u }{ (u+4)(u+2) } du Then...
  49. C

    How to do the indefinite integral

    Homework Statement I hope someone can help me with this integral. It is really an improper integral, but I cannot figure out how to do the indefinite integral on it: integral[e1/x/x3]. Any help would be appreciated, thanks. Homework Equations ? The Attempt at a Solution I...
  50. K

    Solving the Hard Indefinite Integral: e^(3x) * sqrt(1+e^(2x))

    1. Homework Statement [/b] \int e^{3x}\sqrt{1+e^{2x}dx Homework Equations Substitution Parts of Integration The Attempt at a Solution Started off using U substitution setting ex = to u. Then tried to use parts of integration. Now I am stuck.
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