# Boost Your Skills in Finding Antiderivatives

Recent questions in Antiderivatives
Cierra Mclaughlin 2023-02-12

## The antiderivative of cos x is. A)sin x B)cos x C)-sin x D) tan x

Omari Mcclure 2023-02-01

## What is the antiderivative of cos x?A) sin xB) cos xC)- sin xD) tan x

Taniyah Hartman 2023-01-14

## Let F(x) be an antiderivative of $\frac{2{\left(\mathrm{ln}x\right)}^{4}}{3x}$, If F(2)=0, then F(8)=?

Taylor Barron 2022-11-18

## Antiderivative of a function arised in KdV equationI am computing the third non-trivial conservation law of KdV equation ${u}_{x}+6u{u}_{x}+{u}_{xxx}=0$ via the power series expansion method (Here we consider real-valued solutions only).I was given an equivalent form of the PDE:${\left(2{u}^{3}+5{u}_{x}^{2}\right)}_{t}+\left(36{u}^{3}{u}_{x}+6{u}^{2}{u}_{xxx}+10{u}_{x}{u}_{xxxx}+60{u}_{x}^{3}+60u{u}_{x}{u}_{xx}\right)$To finish the job one needs to express$\left(36{u}^{3}{u}_{x}+6{u}^{2}{u}_{xxx}+10{u}_{x}{u}_{xxxx}+60{u}_{x}^{3}+60u{u}_{x}{u}_{xx}\right)$ in a form $\left(\cdots {\right)}_{x}$It is clear that $\left(9{u}^{4}{\right)}_{x}$ is an antiderivative of $36{u}^{3}{u}_{x}$, but what is the antiderivative of$6{u}^{2}{u}_{xxx}+10{u}_{x}{u}_{xxxx}+60{u}_{x}^{3}+60u{u}_{x}{u}_{xx}$ in terms of derivatives of u?

Barrett Osborn 2022-11-03

## Clarification about the Antiderivative of ${x}^{-1/3}$.In some textbook the antiderivative of ${x}^{-1/3}$ is written as$\int {x}^{-1/3}\mathrm{d}x=\frac{3}{2}{x}^{2/3}+C,$where C is a constant. But should not the following function also be considered as an antiderivative of ${x}^{-1/3}$?When ${C}_{0}\ne {C}_{1}$, F(x) cannot be written as $\frac{3}{2}{x}^{2/3}+C$.Edit: I would like to clarify: which one of the following should be the correct answer to $\int {x}^{-1/3}\mathrm{d}x$:- $\frac{3}{2}{x}^{2/3}+C$- or

Kendrick Finley 2022-10-18

## Antiderivative of: $t↦\left(1-{t}^{2}{\right)}^{\lambda }$I would like to find an antiderivative of the function $t↦\left(1-{t}^{2}{\right)}^{\lambda }$ where $\lambda \in {\mathbb{R}}_{>0}$.I really don't know how to proceed. One idea is to use the generalized binomial theorem to get:$\left(1-{t}^{2}{\right)}^{\lambda }=\sum _{k=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{\lambda }{k}\right)\left(-1{\right)}^{k}{t}^{2k}$And by termwise integration I get that a possible antiderivative is $\sum _{k=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{\lambda }{k}\right)\left(-1{\right)}^{k}\frac{{t}^{2k+1}}{2k+1}$.The problem is that this form isn't really helpful. So is there a close form of this? So that I can study the behavior of the function when $\lambda \to \mathrm{\infty }$ on [0,1], for example.

podvelkaj8 2022-10-18

## Equality of two trigonometric integrals on [0,1]I need to show, that:But frankly I cannot see way to solve it. The right-side integral is improper and as far I know both of them don't have the elementary antiderivatives.

joyoshibb 2022-08-10

## Verifying Fourier series of antiderivative of a functionThe function is $f\left(x\right)=\left\{\begin{array}{ll}0,& -\pi \le x\le 0\\ x,& 0.The Fourier series of the function is:$\frac{1}{2}\left(\frac{\pi }{2}\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n}-1}{\pi {n}^{2}}\mathrm{cos}\left(nx\right)+\frac{\left(-1{\right)}^{n+1}}{n}\mathrm{sin}\left(nx\right)$.Since the function can be integrated termwise, the Fourier series for the antiderivative ${\int }_{-\pi }^{x}f\left(x\right)\phantom{\rule{thinmathspace}{0ex}}dx$ is:$\frac{1}{2}\left(\frac{\pi }{2}\right)\left(x+\pi \right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}\left(\frac{\left(-1{\right)}^{n}-1}{\pi {n}^{2}}\mathrm{sin}\left(nx\right)+\frac{\left(-1{\right)}^{n}}{n}\left(\mathrm{cos}\left(nx\right)-\left(-1{\right)}^{n}\right)\right)$The antiderivative of the function is $F\left(x\right)=\left\{\begin{array}{ll}0,& -\pi \le x\le 0\\ \frac{{x}^{2}}{2},& 0. The Fourier series of the antiderivative is:$\frac{1}{2}\left(\frac{{\pi }^{2}}{6}\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}\left(\left(\frac{\left(-1{\right)}^{n}-1}{\pi {n}^{2}}-\left(-1{\right)}^{n}\frac{\pi }{2}\right)\mathrm{sin}\left(nx\right)+\frac{\left(-1{\right)}^{n}}{n}\mathrm{cos}\left(nx\right)\right)$Where I am missing? Why I am having wrong answer. Even if I did some mistakes in calculations, what's with x term obtained in piece wise integration. Thanks in advance.

Max Macias 2022-08-06

## Finding the value of b so ${\int }_{1}^{b}\left(x-2{\right)}^{3}dx=0$.Please how do find $b>1$ so that This question is on a chapter dealing with antiderivatives and I'm not sure how to go about it. At this point it is assumed that I don't know how to integrate yet. I'm also not allowed to use the fundamental theorem of calculus.

wendi1019gt 2022-08-04

## A ball with mass 0.15kg is thrown upwardswith initial velocity 20 m/sec from the roof of a building 30mhigh. There is a force due to air resistance of ${v}^{2}/1325$.where the velocity is measured in m/sec.(a) Find max height above ground ball reaches.(b) Find the time that the ball hits the ground.

comAttitRize8 2022-07-20

## Find the antiderivative of $\sqrt{3x-1}dx$.I got $\frac{2}{3}\left(3x-1{\right)}^{3/2}+c$ but my book is saying $\frac{2}{9}\left(3x-1{\right)}^{3/2}+c$.Can some one please tell me where the 2/9 comes from?

Alexandra Richardson 2022-07-17

## How to prove the function f has an antiderivative?There is a question in our analysis book and I have lots of problem with it. It says that:set $f\left(x\right)=\left\{\begin{array}{ll}0\phantom{\rule{1em}{0ex}}& x\le 0\\ \mathrm{sin}\left(\frac{\pi }{x}\right)\phantom{\rule{1em}{0ex}}& x>0\end{array}$$g\left(x\right)=\left\{\begin{array}{ll}0\phantom{\rule{1em}{0ex}}& x\le 0\\ 1\phantom{\rule{1em}{0ex}}& x>0.\end{array}$prove that f has an antiderivative but g does not.My first problem is about antiderivative of g. I think it has an antiderivative and it is$\begin{array}{rl}G\left(x\right)=& \left\{\begin{array}{ll}0\phantom{\rule{1em}{0ex}}& x\le 0\\ x\phantom{\rule{1em}{0ex}}& x>0.\end{array}\end{array}$Why we can't say that G is antiderivative of g? My second problem is about finding the antiderivative of f. As you may know antiderivative of $\mathrm{sin}\left(\frac{\pi }{x}\right)$ can not be shown by the elementary functions. So, to prove that f has antiderivative I can't find a function like F which its derivative is f and I need to use another approach to prove it, but I don't have any idea about what should I do?

slijmigrd 2022-07-15

## Order of antiderivatives of Schwartz functionsFor a Schwartz function $f\in \mathcal{S}\left(\mathbb{R}\right)$ it is known that$\mathrm{\exists }F\in \mathcal{S}\left(\mathbb{R}\right):{F}^{\prime }=f\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f\left(x\right)dx=0$in this case $F\left(x\right)={\int }_{x}^{+\mathrm{\infty }}f\left(t\right)dt$, see related question here.Question: Does there exist $f\in \mathcal{S}\left(\mathbb{R}\right)$ with ${f}^{\left(-k\right)}\in \mathcal{S}\left(\mathbb{R}\right)$ for all $k\ge 0$ (my guess would be no)I.e. for a given f we can test if it has an antiderivate F in the Schwartz space by simply checking if its mean is 0. Then we can do the same with F. Does this process always stop at some point?My work: It seems that all functions of the type $f\left(x\right)=p\left(x\right){e}^{-\alpha {x}^{2}}$ with p polynomial only have antiderivatives of finite order in $\mathcal{S}\left(\mathbb{R}\right)$. (I think I can prove this). As one can construct an orthonormal basis of $\mathcal{S}\left(\mathbb{R}\right)$ with such functions, e.g. the Hermite functions, I would guess that there is no function with antiderivates of arbitrary order in $\mathcal{S}\left(\mathbb{R}\right)$.Remark: If there is such an f, then $\left({x}^{k}\star f\right)=0$ for all $k\ge 0$; then via Fourier transform, this would imply that in the sense of distributions:$0=\mathcal{F}\left[{x}^{k}\star f\right]=\mathcal{F}\left[{x}^{k}\right]\mathcal{F}\left[f\right]=\left(\frac{i}{2\pi }{\right)}^{k}{\delta }^{\left(k\right)}\left(w\right)\stackrel{^}{f}\left(w\right)$

auto23652im 2022-07-14

## The extension of this derivative notation ${f}^{\left(0\right)}=f,{f}^{\left(1\right)}={f}^{\prime },...$Just a quick question on this notation, is ${f}^{\left(-1\right)}$ used for antiderivatives?

Lena Bell 2022-07-13

## Is there any elementary function whose antiderivative contains an exact constant?Let's say we have . Now it's obvious to me why F is the class of functions whose derivative yields f(x). However, I was curious if it is possible for the antiderivative to be something such as $F\left(x\right)=\mathrm{tan}\left(x\right)+5+C$ as an example. I can see how we would merge this to become just $F\left(x\right)=\mathrm{tan}\left(x\right)+C$ but I'm wondering if it is possible for an elementary antiderivative to contain a constant that the entire class of antiderivatives share. My feeling is that this would never happen but I can't seem to figure out exactly why.

Jamison Rios 2022-07-10

## Continuous complex function without antiderivativeIt's a well-known result that every real continuos function has an antiderivative. Is this theorem still true for a complex function? If not, can someone point out a counter-example (and proof that it is indeed a counter-example)?

dream13rxs 2022-07-10

## Antiderivatives - part c, solving for x$MR\left(x\right)=4x\left({x}^{2}+26,000{\right)}^{-2/3}$I'm already lost at the part $2\int {u}^{-2/3}$. How did they get $6{u}^{1/3}+C$.a)Find the revenue functionb) What is the revenue from selling 250 ​gadgets?​c) How many gadgets must be sold for a revenue of at least ​\$50​,000?Solve for x. (How?)$6\left({x}^{2}+26,000{\right)}^{1/3}-150=50$

Dayanara Terry 2022-07-09