Questions tagged [indefinite-integrals]

Question about finding the primitives of a given function, whether or not elementary.

Filter by
Sorted by
Tagged with
0
votes
0answers
17 views

Euler Substitution For Logarithm Examples

How do I solve integrals of the form $$ \left(\sqrt{ax^2 + bx + c}\right)\left(ln(\sqrt{ax^2 + bx + c}) \right)$$ using euler's substitution. Tried finding examples online but could not find one. ...
7
votes
2answers
122 views

$\int{x^k dx}$ as $k \rightarrow -1$ "paradox"

While I was studying integrals by my own, I learnt these two rules for integrating $f(x) = x^k$: if $k \neq -1$, then $\int{x^k dx}=\frac {x^{k+1}}{k+1}+c$; if $k =- 1$, then $\int{x^{-1} dx} = \ln {|...
0
votes
1answer
34 views

Separable ODE - Integral involving Elliptic integral

I am trying to solve the equation $$ \dot y(t) = -\sqrt{ y(t)^6 + y(t)^2 + a } $$ with $a \leq y(t)^6+y(t)^2$ for all $t\geq 0$. This is a first order separable ODE. According to Wolfram Alpha the ...
0
votes
1answer
69 views

Integral of $1/(ax^2 +bx +c)^n$

How can we obtain the recursion relation for the integral of the following rational function? $$ \begin{align} \int \frac{dx}{(ax^2 +bx +c)^n} &= \frac{ 2ax+b }{ (n-1) (4ac-b^2) (ax^2 +bx +c)^{n-...
0
votes
2answers
42 views

Improper integral with residue theorem why is condition no real zeros needed?

If I apply the residue theorem to evaluate the improper integral $\int_{-\infty}^{\infty}\frac{1}{x^2}\mathrm{d}x$ then all necessary conditions are fulfilled except that f has no real singularities. ...
1
vote
1answer
73 views

Euler Integral Substitution

I was trying to find the indefinite integral for this function, I have tried integration by parts but did not make much headway, and so I was wondering if Euler substitution could be used in a way to ...
3
votes
3answers
118 views

What is the difference between the below notations?

I've often come across two ways of integrating which I think mean the same thing. $\displaystyle \int f(x) dx$ $\displaystyle \int_{-\infty}^\infty f(x) dx$ Do these two mean the same thing because, ...
1
vote
1answer
53 views

Limit of an indefinite integral

I need to prove that this limit exists and calculate it. I tried integration by parts, resulting with an integral which is called "The Cosine Integral", denoted $Ci(x)$. However, I need to ...
4
votes
1answer
70 views

Running through math confusion how to finish this integral using Beta function.

I am working on this easy integral but I want to use Beta Trigonometric Function to solve it. I know the answer is $\pi$ by using the half angle formula and help from u-substitution. $\int_{0}^{2\pi} \...
-1
votes
5answers
81 views

When integrating, is $C=0$ a unique solution?

Why do we treat a certain $C$ as adding nothing? take $\sin(x)$. It's usually agreed that: $$\int\sin(x)dx=-\cos(x)+C$$ But I could define a function like: $$\mathrm{sock}(x) = \cos(x)-1$$ And ...
5
votes
2answers
364 views

Closed form of definite integral from 2006 MIT Integration Bee

The judges claim that the closed form of $$\int_0^\infty\frac{e^{3x}-e^x}{x(e^{3x}+1)(e^x+1)}\text{d}x$$ is $\frac{\text{log}(3)}{2},$ although no internet calculator except Wolfram|Alpha finds a ...
1
vote
5answers
222 views

(Disagreement among reputable users) Indefinite integral vs. Definite integral vs. Anti-derivative

Suppose, I have a function $\cos(x)$. Now, $$\int{\cos(x)dx}$$ $$\sin(x)+c\\ {\text{[c is a constant]}}$$ Now, there could be an infinite number of values for $c$. For example, $c=1,2,-2,\pi,-\pi, 0, \...
2
votes
0answers
61 views

Evaluating $\int_{1}^{\infty}\ \frac{\sqrt[3]{x-1}}{\sqrt{x^3+3}}dx$

I tried doing : $\int_{1}^{\infty}\ \frac{\sqrt[3]{x-1}}{\sqrt{x^3+3}}dx \approx \int_{1}^{\infty}\ \frac{x^\frac{1}{3}}{x^\frac{3}{2}}dx = \int_{1}^{\infty}\ \frac{1}{x^\frac{7}{6}}dx $ According ...
0
votes
0answers
59 views

Stuck solving this indefinite integral

Problem: Solve integral: Approaches I've tried: Solving by parts led me nowhere but to more confusion (maybe I've taken the wrong parts?): Trying to replace $1+x$ or $1-x$ with another variable led ...
-1
votes
1answer
92 views

calculate :$\int_0^\infty\ e^{-a^2(y^2+\frac{p^2}{4a^2y^2})}\,dy$ [duplicate]

Trying to calculate $$\int_0^\infty\ e^{-a^2(y^2+\frac{p^2}{4a^2y^2})}dy.$$ When I searched it on Google, I found the formula $$\int_{-\infty}^{\infty} \ e^{-a(x+b)^{2}}dx=\sqrt\frac{\pi}{a},$$ but ...
1
vote
1answer
99 views

How to solve this indefinite integral? $\int {t}{\sqrt{1+\cos(t)}} \,dt$

I tried solving the following integral: $\int {(t+\sin(t))}{\sqrt{1+\cos(t)}} \,dt$ I split the problem in two parts. The first part gave me this: $\int {\sin(t)}{\sqrt{1+\cos(t)}} \,dt = -\frac{2}{3}{...
3
votes
2answers
60 views

Trying to evaluate : $\int \frac{e^{x^2}+2xe^{x}}{e^{x^2}+e^x} dx$

Here's my approach : $$\int \frac{e^{x^2}+2xe^{x}}{e^{x^2}+e^x} dx = \int\sum\limits_{n=0}^{\infty} \frac{\frac{(x^2)^n}{n!}+\frac{2x^{n+1}}{n!}}{\frac{(x^2)^n}{n!}+\frac{x^n}{n!}} dx = \int\sum\...
1
vote
4answers
144 views

Integral of $\int \frac{\sqrt{x^2-1}}{x}dx$

Evaluate the integral of $$\int \frac{\sqrt{x^2-1}}{x}dx$$ Attempt: I've tried taking $x=\sec y$, $$ \int \frac{\sqrt{\sec^2y-1}}{\sec y}dx $$ How to proceed further?
0
votes
2answers
80 views

Why am I getting two different answers while calculating $\int \frac{dx}{\sqrt{2ax-x^2}}$?

I was calculating the antiderivative of the function $\frac{1}{\sqrt{2ax-x^2}}$. I got two different answers in two different ways. First way: $$\int \frac{dx}{\sqrt{2ax-x^2}}=\int \frac{dx}{\sqrt x\...
0
votes
2answers
70 views

Question on integral containing exponential and sine function

How to get estimate on following integral: $$f(x)=\int\frac{\sin^2(x)}{e^{\sin^2(x)}}dx \,?$$ I tried doing it so by putting $u= \sin^2(x)$, that way we get: $$\int\frac{\sqrt{u}}{\sqrt{1-u^2}e^u}dx.$$...
2
votes
2answers
99 views

Finding the antiderivatives of a trigonometric function in two different intervals

I wanted to find the antiderivatives of the function $f(x)=\frac{1}{sin(x)+cos(x)+2}$ in $[0,\pi[$ first and then in $[0,2\pi]$. Now, as long as the first set is concerned, I used Weierstrass's ...
4
votes
2answers
138 views

How many ways to deal with the integral $\int \frac{d x}{1-\sin x \cos x}$?

Multiplying both numerator and denominator of the integrand by $\sec^2 x$ yields \begin{aligned} & \int \frac{d x}{1-\sin x \cos x} \\ =& \int \frac{\sec ^{2} x}{\sec ^{2} x-\tan x} d x \\ =&...
4
votes
4answers
194 views

How many ways to deal with the integral $\int \frac{d x}{\sqrt{1+x}-\sqrt{1-x}}$?

I tackle the integral by rationalization on the integrand first. $$ \frac{1}{\sqrt{1+x}-\sqrt{1-x}}=\frac{\sqrt{1+x}+\sqrt{1-x}}{2 x} $$ Then splitting into two simpler integrals yields $$ \int \frac{...
0
votes
1answer
66 views

How is $\frac{\sin^2x}{\cos^2x+1}=\frac{\tan^2x}{1+\sec^2x}$?

I am reading my Calculus material and they present me this: $$ \int\frac{\sin^2x}{\cos^2x+1}dx=\int\frac{\tan^2x}{1+\sec^2x}dx $$ I tried around playing with trig identities but I can't reach this ...
0
votes
1answer
53 views

For non negative integers $\sum{P\left(X\geq x\right)}=\int{\left(1-F\left(x\right)\right)\mathrm{d}x}$

I want to show that for non negative integers $X$ we have $$\sum_{x=1}^{\infty}{P\left(X\geq x\right)}=\int_{0}^{\infty}{\left(1-F\left(x\right)\right)\mathrm{d}x}$$ I know that $P\left(X\geq x\right)=...
0
votes
0answers
34 views

How can I choose or calculate the function $f(t)$ in $(a*t+b)/(c*f(t)+d)$ to best approximate a given linear function in the t=[0,1] interval?

I have an expression with an unknown function $p(t)$ $$g(t)=(921600*(-1333.33340000000*t + 2000)/(-3.36000004800000e6*p(t) + 3840000)$$ I would like to calculate somehow the $p(t)$ function that the $...
1
vote
1answer
64 views

Evaluate the following integral: $ \int \frac{1+x^2 \ln x}{x+x^2 \ln x}dx $ [duplicate]

I've just learnt integration recently and I'm having trouble where to start solving it. Evidently this can't be solved with direct substitution (atleast in my knowledge) and I think I'm supposed to ...
-1
votes
2answers
85 views

Trigonometric anti derivative [closed]

The question is the top integral ,I did substitution of 1+cos ,x+sin2x,sin,cos all of these not works, and I separate them out into two pieces and I am stuck in a new antiderivative :x /(1+cosx)² \...
4
votes
2answers
110 views

How to solve a complicated ODE

The equation is $$ f^{\prime}(x) = \gamma \frac{f(x)+f^2(x)}{\log\left(\frac{f(x)}{1+f(x)}\right)} $$ with the initial condition $f(0)$, where $x\geq 0$ and $f(0)\geq0$. The solution is $$f(x)=\frac{1}...
0
votes
0answers
73 views

What is the integral of $1/x$ on the entire complex plane?

Usually, when one asks for the integral of $1/x$, they’ll be answered with “$ln(x)$”, but it’s “undefined” for negative values. Sometimes, they’ll be answered with “$ln(|x|)$”, which is defined in the ...
0
votes
0answers
70 views

Solving $xy’’’+(1-m)y’’+2y=0,y=y(x)$ to prove hypergeometric representation of Abramowitz function from NIST.gov

Intro: I was looking for a fresh new problem to solve and was inspired by Evaluation the Elsasser function:$$\text E(y,u)=\int_{-\frac12}^\frac12e^{\frac{2\pi uy\sinh(2\pi y )}{\cos(2\pi x)-\cosh(2\...
1
vote
3answers
56 views

Trying to evaluate $\int \frac{1}{\sin(x)\cos^3(x)} \,dx$ and got stuck

So, I am trying to evaluate the following anti-derivative: $$\int \frac{1}{\sin(x)\cos^3(x)} \,dx$$ I reached a point where I have the following: $$\int \frac{\sin(x)}{\cos^3(x)} + \frac{1}{\sin(x)\...
4
votes
2answers
73 views

I have an equation derived from Ising model for the correlation distance. Is there a way to rewrite this integral to make in manageable?

It took a lot of work to get here, but now I am stuck. I want to show that $$C(\vec{r}) = \int \mathrm{d}^d k { e^{ik \cdot \vec{r}}\over t+ k^2}$$ evaluates to $$C(\vec{r}) \approx \Bigl(\frac{\sqrt{...
3
votes
1answer
61 views

Calculating integrals of the form $\int\frac{dx}{(1+x^{q})^\frac{p}{r}}$

I am trying to understand how to calculate integrals of the form $\int\frac{dx}{(1+x^{q})^\frac{p}{r}}$ , where $p,q,r \in \Bbb Z$. I know how to calculate the integral $\int \frac{dx}{(1+x^2)^\frac{...
0
votes
1answer
130 views

Evaluate $\int _{ }^{ }\frac{1}{\sqrt{1+x^3}}dx$

My attempt at solution is... $ \int _{ }^{ }\frac{1}{\sqrt{1+x^3}}dx\ \left(u=x^3,\ dx=\frac{du}{3x^2}\right) $ $ \int _{ }^{ }\frac{1}{3x^2\sqrt{1+u}}du\ \ \left[\sqrt[3]{u}=x\right] $ $ \int _{ }^{ }...
4
votes
1answer
61 views

How to solve this nonlinear diff eq of celestial mechanics?

$$(\dot{r})^2 = \frac{2 \mu}{r} + 2h$$ Where mu and h are constants. I have no idea how to solve it, maybe there is a trick I didn't know. The only thing that came in mind is to integrate $$\int \frac{...
4
votes
2answers
110 views

Evaluating an improper integral using the Residue Theorem

I have: \begin{equation} \int_{-\infty}^{\infty}\frac{x^2}{(x^2+1)(x^2+9)}dx \end{equation} and I want to solve it using a complex closed contour on C. I do the following: \begin{equation} \int_{-\...
1
vote
3answers
87 views

The function $f(x) = x^3-4a^2x$ has primitive function $F(x)$ . Find the constant $a$ so that $F(x)$ has the minimum value $0$ and $F(2)=4$.

The function $f(x) = x^3-4a^2x$ has primitive function $F(x)$ . Find the constant $a$ so that $F(x)$ has the minimum value $0$ and $F(2)=4$. This is what I tried below. I am not sure how to use the ...
11
votes
1answer
354 views

Taking constants out of indefinite integrals

In the case of definite integrals, the linearity property implies that constants can be taken out of the integrals, $$\int_{a}^{b} \alpha f(x) d x=\alpha \int_{a}^{b} f(x) d x \tag{1}$$ However, in ...
0
votes
1answer
42 views

How do I find $\frac{\mathrm{d}}{\mathrm{d}x} \int_{x}^{\infty} f(t, x) \mathrm{d} t$?

I am trying to compute the following derivative: $$\frac{\mathrm{d}\Big(\int_{x}^{\infty} f(t, x) \mathrm{d} t\Big)}{\mathrm{d}x} \text{.}\tag{1}$$ It is straightforward to compute the following ...
4
votes
0answers
45 views

Explicit elementary primitive (integral) of $x / \sqrt{P(x)}$ and Galois group of $P$

I am reading some stuff on Risch's algorithm here (Wikipedia in French), about finding explicitly some primitive of functions in terms of "elementary functions" (composition of polynomials, ...
1
vote
0answers
16 views

Given the time-limited continuous function $q(t) = \int_{-\infty}^t h(u)\,du$, Could $h(t)$ be discontinuous? Has $h(t)$ to be of compact-supported?

Given the time-limited continuous function $q(t) = \int_{-\infty}^t h(u)\,du$, Could $h(t)$ be discontinuous? Is mandatory for $h(t)$ to be a function of compact-support? (so, also time-limited?) ...
1
vote
1answer
125 views

Integral of $1/\cos^2 x$

Usually, I say to students that $\int \frac {1}{\cos^2 x}\,dx=\int \sec^2 x=\tan x +c$ based directly on the list of immediate integrals. The other day a student asked me if we can evaluate the ...
2
votes
0answers
57 views

Integral involving Airy functions and exponential

I'm struggling calculating the following integral: $$\int_ {-\infty}^\infty \operatorname{Ai}(x)~\operatorname{Ai}(x-2it) e^{2(x-it)} dx $$ Where x is a complex number. I'm studying "Airy ...
0
votes
0answers
27 views

Integral representation of modified Bessel function $K_{\frac{1}{4}}$

I am looking for dereviation of integral representation of modified Bessel function, such that: $$ \int_{-\infty}^{+\infty} dq e^{- \frac{m^2}{2}q^2 - \frac{g}{4!} q^4} = \frac{2}{m} \sqrt{\frac{3m^4}{...
1
vote
1answer
56 views

Simplifying the integral $\int \frac{x^2-1}{x^2+1}\cdot \frac{1}{\sqrt{1+x^4}}dx$.

I have the following integral that I must simplify by making two substitutions \begin{align} \int \frac{x^2-1}{x^2+1}\cdot \frac{1}{\sqrt{1+x^4}}dx, \end{align} one of them is \begin{align} y = \frac{...
1
vote
0answers
85 views

Is there an antiderivative for $e^{-\left( x + \frac{1}{x}\right)}$?

I was playing around with some integrals I made up myself and was trying to find a closed-form for $$ \int_{0}^{t} e^{-\left( x + \frac{1}{x}\right)} \ dx, \qquad t < \infty $$ I'm aware that if ...
1
vote
0answers
59 views

Inverse Function Integral

$$\int \frac{1}{x\sqrt{x^2-25}} \,dx$$ So I realized that there are multiple ways to solve the question above: I could use the inverse secant identity: $\int \frac{1}{x\sqrt{x^2-1}} \, dx$, giving ...
1
vote
1answer
108 views

Solve indefinite integral $\int\frac{x^2}{1-x^2+\sqrt{1-x^2}}dx$

$$\int\frac{x^2}{1-x^2+\sqrt{1-x^2}}dx$$ I multiply the integral so that I can get $-x^2$ in the numerator. I then expand the fraction so I can split the integral into easier integrals. $$-\int\frac{-...
0
votes
2answers
35 views

Integration by Substitution for $\int \left ( \frac{dx}{\sqrt[]{a^{2}-x^{2}}} \right )$ gives two results ? Which is correct and why?

Just applying Integration by Substitution for the given equation (Method#1 & Method#2), Let, $$F(x)=\int \left ( \frac{dx}{\sqrt[]{a^{2}-x^{2}}} \right )\\\tag{1}$$ $\underline{Method \ No.\ 1:}$ ...

1
2 3 4 5
96