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Questions tagged [indefinite-integrals]

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

-5
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0answers
34 views

How to calculate increasing and decreasing with integration. [on hold]

F(x)=d/dx (integral from 0-x^5 of 2t^3dt) The question is to find F(1). To find the normal function, you solve for what's under the integral because F=(integral)...
3
votes
2answers
65 views

Evaluate $\int_0^1 \frac{1}{\sqrt x+\sqrt {(1-x)}}dx$

Evaluate $I=\int_0^1 \frac{1}{\sqrt x+\sqrt {(1-x)}}dx$. I applied $x=\sin^2\theta$,that makes $I=\int_0^{\pi/2} \frac{\sin2\theta}{\sin\theta+\cos\theta}d\theta$,but the further proceedings makes $I$...
0
votes
2answers
49 views

Integrate $\int \frac{\cos^2 x}{(1 - \cos x)\sin x} dx$

Integrate $\int \frac{\cos^2{x}}{(1 - \cos{x})\sin x} dx$ So far I have gotten here: $-\int -\frac{\cos^2 (x) \sin x}{(\cos (x) - 1)(\cos^2 (x)-1)} dx$ here I can substitute $u = \cos x$ , $ -\...
2
votes
2answers
27 views

Evaluate $\int \frac{ 2\exp\left((-\tan^2(t))/a^2\right) }{\cos^3(t)a^2}dt$ using substitution.

Evaluate : $\displaystyle\int \frac{ 2\exp\left((-\tan^2(t))/a^2\right) }{\cos^3(t)a^2}dt$ The hint was to use $x=\cos(t)$ and the fact that $\int f'(x)e^f(x)dx=e^f(x)$. Since $$x=\cos(t)\;\text{...
4
votes
1answer
156 views

Integration of $\int^{1}_{-1} \frac {1}{3} \sinh^{-1} \left( \frac {3\sqrt 3}{2} (1-t^2) \right) dt$

Recently I came across with respect to this post of mine hyperbolic solution to the cubic equation for one real root given by $$ t=-2\sqrt \frac {p}{3} \sinh \left( \frac {1}{3} \sinh^{-1} \left( \...
0
votes
0answers
18 views

Bessel function limit

I was doing a physics problem and encountered with following limit of Bessel function : $\lim_{R\to\infty} R^2J_n(\lambda R)$ and $\lambda = \sqrt{\omega^2 - \frac{1}{R^2}}$ I got this limit from ...
2
votes
3answers
190 views

Integrate $\int\frac{\sin^{-1} (x)}{(1-x^2)^{3/4}} \,\mathrm d x$

Integrate $$\int\frac{\sin^{-1} (x)}{(1-x^2)^{\frac{3}{4}}} \,\mathrm d x$$ I have followed some steps from here, but am not able to solve this question. Any help would be appreciated. Update: After ...
0
votes
1answer
15 views

Partial fraction decomposition with two variables

While I was solving some exercises as training for a test I'm gonna have, I've noticed that some integrals that I have to solve, I don't know how to do them, and I have no explanation on my papers and ...
1
vote
1answer
86 views

Find $\int \frac{x^n\ln x}{(x^{n+1}+1)^n}dx$,

Find $\displaystyle\int \dfrac{x^n\ln x}{(x^{n+1}+1)^n}dx$, where $n\in\mathbb{N}$, $x\in(0,\infty)$ I think I saw this on mathstack, but I can't find it. Anyway, for this problem, my idea was to ...
2
votes
2answers
55 views

Ways to simplify arctan() in integral results?

Lately when I was computing $$\int\frac{\mathrm{d}x}{1+\sqrt{1-2x-x^2}},$$I got the result$$-2\arctan{\frac{\sqrt{1-2x-x^2}-1}{x}}-\ln\left(1-\frac{\sqrt{1-2x-x^2}-1}{x}\right)+\mathbf{C}.$$However, ...
-4
votes
1answer
38 views

Integrate $\int\frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}\,dx$ [closed]

Can some one please help me with this integral $$\int\frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}\,dx$$
7
votes
2answers
125 views

Is $e^{\int \frac {1}{x}dx}$ equal to $x$ or $|x|$?

I encountered this expression quite a lot of times as a part of the integrating factor while solving linear differential equations. $$e^{\int \frac {1}{x}dx}$$ For sometime, I wrote it as $x$, and ...
0
votes
2answers
40 views

Integrate $\int \frac{\sqrt{9x^2-1}}{2x}dx$

What is $\int \frac{\sqrt{9x^2-1}}{2x}dx$? I tried to form a triangle with $\cos\theta=\frac{1}{3x}$ and $\sin\theta=\frac{\sqrt{9x^2-1}}{3x}$ to use as substitution. But I can't get rid of all the $...
0
votes
0answers
35 views

How to proof $\int_{-\infty}^{+\infty}\sin(w_1*t)\sin(w_2*t)\,dt = 0$ if $w_1 \neq w_2$

In my math script (signal theory) it says that two functions are orthogonal to each other when $\int_{-\infty}^{+\infty}s^\star(t)u(t)\,dt = 0$. Now I want to prove that $$\int_{-\infty}^{+\infty}\sin(...
0
votes
0answers
42 views

What is the default definition of $F(x)$ given $f(x)$?

To avoid the confusing (to me) notation, let's call $g(x)$ the antiderivative of $f(x)$, such that $g'(x) = f(x)$. I've seen $F(c)$ defined as the definite integral of $f(x)$ from $0$ to $c$ with ...
4
votes
1answer
71 views

Is this integral unsolvable?

So I took an integration test in AP Calculus yesterday and everything went smoothly except for one question. $$\int \frac{e^x}{x^2}dx$$ I tried chain rule, $u$ substitution, and all methods we have ...
1
vote
2answers
49 views

Indefinite integral $\int \frac{(1-x)^3}{x \sqrt[3]{x}}$

Find the indefinite integral $\int \frac{(1-x)^3}{x \sqrt[3]{x}}$: I guess the fay forward would be to find a suitable substitution, but I am struggling with that.
1
vote
3answers
40 views

Simple integral with $e^x$ - how to decompose it?

How to calculate this integral? $$\int \frac{e^{2x}+2}{e^x+1}dx$$ I have tried various substitions such as: $t = e^x, t = e^x + 1, t = e^x +2, t = e^{2x}$ and none seem to work. According to ...
0
votes
1answer
67 views

Anti-derivative of $\frac{\exp(x)-1}{x}$

I am looking for the antiderivative of $$\frac{\exp(x)-1}{x}$$ I showed that it is equivalent to calculate $$\sum_{n=1}^{\infty}\frac1n \frac{x^n}{n!}$$ but I can't find both of the solutions. If ...
0
votes
2answers
36 views

Indefinite integral of rational polynomial function

How do I integrate $\int \frac {(x-3)}{(x^2-2x+4)^2} dx $ I know that I have to integrate via recursion. But doing so I get to a point where I have to find $\int \frac {x^2}{(x^2-2x+4)^2} dx $ which ...
0
votes
1answer
43 views

Calculating an infinite integral of log-normal distribution

The integral is: $\int^\infty_0 x \exp{\Big(\frac{-(\log{x}-\mu)^2}{2\sigma^2}\Big)}dx \ \ \ $ (it is a second moment of log-normal distribution). I've tried several subsitutions, such as $u=\log{...
0
votes
1answer
25 views

Primitive of a composite function

I'm reading Zorich, Mathematical Analysis I, and I found a not clear step in the paragraph on Primitives. The particular sentence is shown below (adapted). From the definition of primitive of a ...
1
vote
3answers
37 views

Finding $\int\frac{\sin x+\tan x}{\cos x+\csc x}dx$

Finding $\displaystyle \int\frac{\sin x+\tan x}{\cos x+\csc x}dx$ what i try $\displaystyle \Lambda =\int\frac{\sin^2 x(1+\cos x)}{\cos x(\sin x\cos x+1)}dx$ $\displaystyle \Lambda=\int\frac{\sin^4 ...
1
vote
2answers
52 views

Can some improper integrals be directly evaluated?

Consider the following improper integral: $\displaystyle \lim\limits_{\delta{r} \to 0} \int^r_{\delta r} f(r) dr \tag{1}$ where $f(r)$ is finite everywhere and $f(0)=$ not defined. Then, will ...
4
votes
1answer
83 views

$\int\frac{dx}{x(x+1)(x+2)\cdot\space…\space\cdot(x+n)}$ [duplicate]

I've been trying to solve explicitly the following indefinite integral: $$\int\frac{dx}{x(x+1)(x+2)\cdot\space...\space\cdot(x+n)}$$ I tried to perform partial fraction decomposition, and after ...
1
vote
2answers
72 views

Calculate the integral $\int \frac{2-3x}{2+3x} \sqrt{\frac{1+x}{1-x}}dx$ [duplicate]

I have to calculate the integral $\int \frac{2-3x}{2+3x} \sqrt{\frac{1+x}{1-x}}dx$. I tried the following substitutions: $x \rightarrow \frac{1+t}{1-t}, x \rightarrow \frac{1-t}{1+t}, x \rightarrow \...
0
votes
0answers
25 views

How to integrate Heaviside function multiplied with a function

For example Wolfram|Alpha gave me this result. But I can't understand how do we achieve this result. How do we take indefinite integral of a function multiplied by heaviside function ?
0
votes
2answers
38 views

Any simple integration to this indefinite integral?

$I =\displaystyle\int \dfrac{\sqrt{4+9x^4}}{x^3}dx$ One method we have tried is to use the substitution $x^2=\displaystyle\frac2{3\tan\theta}$ ,but it seems hard to change back the $\theta$ to x in ...
3
votes
1answer
49 views

Find $\int |\sin(x) + \cos(x)|\ dx$

$$\int |\sin(x) + \cos(x)|\ dx$$ Do I just do: $$\operatorname{sgn}(\sin(x) + \cos (x)) \int \sin(x) + \cos(x) \ dx = \frac{\sin(x) + \cos (x)}{|\sin(x) + \cos(x)|} \int \sin(x) + \cos(x)\ dx$$ ...
1
vote
2answers
43 views

Find $\int \frac {1} {(x-a)^n} dx$

Find $\int \frac {1} {(x-a)^n} dx$ where $n \in \mathbb{N}, a \in \mathbb{R}$ Am I supposed to solve this using substitution?
2
votes
1answer
44 views

Evaluating indefinite integrals of the form $\int \frac{x^2 \,dx}{a x^5 + b}$

Evaluate the indefinite integral $$\int \frac{x^2 \,dx}{a x^5 + b},$$ for real parameters $a, b \neq 0$. No apparent substitutions simplify the expression (if the exponent of $x$ were an integral ...
0
votes
2answers
35 views

Primitives on $[a,c)$ and $(c,b]$ implies primitives on $[a,b]$

Let $a<c<b \in \mathbb R$ and $f:[a,b] \to \mathbb R$ be continuous in c. If $f$ has primitives on $[a,c)$ and $(c,b]$ then $f$ has primitives on $[a,b]$. Can somebody help me, please? I have no ...
3
votes
2answers
48 views

Explanation how it can be that $f'(x) = g(x)$ but wolfram alpha says $\int g(x) \neq f(x)$?

I've just started learning about antiderivatives/primitive functions/indefinite integrals, and I have the functions $f(x) = 3 \ln((\frac{x+2}{3})^2 + 1)$ $g(x) = \dfrac{2\frac{x+2}{3}}{(\frac{x+2}{...
-2
votes
1answer
64 views

Integrating this beast $\int\frac{3}{x\sqrt{x^2+9}}dx$ [closed]

How do I integrate this beast $$\int\frac{3}{x\sqrt{x^2+9}}dx?$$ Using the substitution $t = \frac{1}{x}$ (the substitution is given). My attempt: https://imgur.com/YSVXMdI alternative sub: t = a ...
2
votes
1answer
37 views

1st order differential linear equation, question on absolute value

I'm trying to find the general solution to this equation: $$x \frac{dy}{dx}+3(y+x^2)=\frac{\sin(x)}{x} $$ Standard form puts it like this: $$\frac{dy}{dx}+\frac{3}{x}y=\frac{\sin(x)-3x^3}{x^2} $$ To ...
1
vote
5answers
75 views

A mistake on computing $\int \frac{dx}{\sqrt{x+1}+1}$

I have to find $\int \frac{dx}{\sqrt{x+1}+1}$. This was my attempt, which is wrong and I cannot find where exactly is the mistake. First I write $\frac{1}{\sqrt{x+1}+1}=\frac{\sqrt{x+1}-1}{x}=\frac{\...
1
vote
1answer
46 views

Find the value of $\int \frac{x^{n}\operatorname{ln}(x)}{(x^{n+1}+1)^{n}}dx$, where $n $ is any natural number different from $0,1,2$

I had to find the value of $\int \frac{x^{n}\operatorname{ln}(x)}{(x^{n+1}+1)^{n}}dx$, where $n $ is any natural number different from $0,1,2$ and $ x$ is a positive, real number. By a change of ...
1
vote
4answers
88 views

Integrating $\int\frac{dx}{(x-a)(x-b)}$ by means of a trigonometric substitution

I've been trying to integrate $$\int\frac{dx}{(x-a)(x-b)}$$ By using the substitution $$x=a \cos^2 \theta + b \sin^2 \theta$$ The only problem here is I arrived at the result $$\frac{2}{a-b} \ln |\...
4
votes
2answers
52 views

Formulation of Bioche's rules in familiar notation

I was trying to find an interesting problem for my physics students involving a nontrivial flux integral, and I came up with one that produced the integral $$\int \frac{dx}{1+\beta\cos x}$$ ($\beta^...
0
votes
1answer
25 views

Complex Irrational Integration

Calculation of $$\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}dx$$ Try: put $x=t^2$ and $dx = 2tdt$ Let $$I = 2\int\frac{(t^2-1)\sqrt{t^8+2t^6-t^4+2t^2+1}}{t(t^2+1)}dt$$ $$I = 2\int \frac{\...
0
votes
1answer
58 views

How to rationalise an integral $\int \frac{\sqrt{3 + 2x - x^2}}{x + \sqrt{3 + 2x - x^2}}dx$

$3 + 2x - x^2 = - (x + 1)(x - 3)$, so I've already tried the substitution $$t = \sqrt{\frac{-1(x + 1)}{x - 3}}$$ and also $t = \sqrt{3 + 3x - x^2}$. But seems it doesn't work in this task.
3
votes
1answer
38 views

Double integral of $xe^y$ over the area inside $x^2 + y^2 = 1$ but outside $x^2 + y^2 = 2y$

My question goes like this: Let R be the area inside $x^2 + y^2 = 1$ and outside $x^2 + y^2 = 2y$. Calculate $\int\int_R xe^y dA$. How sould I approach this question? I tried to use integration ...
1
vote
0answers
25 views

Indefinite integral involving the product of two generalized Laguerre polynomials

I am trying to find the indefinite integral \begin{align} \int{x^{\alpha +1}e^{-x}\left(L_{m}^{\alpha}(x)\right)^{2}dx} \end{align} where $L_{m}^{\alpha}(x)$ is the generalized Laguerre Polynomial, ...
1
vote
2answers
131 views

Evaluate $\int_{0}^{\pi} x\sin\big(\frac{1}{x}\big)\cos x \,dx$

I wonder if an integral of the form $$\int_{0}^{\pi} x\sin\Bigl(\frac{1}{x}\Bigr)\cos x \,dx$$ which can be further simplified to $$\int_{0}^{\pi} \frac{\sin (x^{-1})}{(x^{-1})}\,\cos x\, dx=\cos x\...
3
votes
0answers
47 views

Is integration by parts used in this equality?

The starting point is this convolution $$ \frac{\partial v_0}{\partial G}(t) = \int_0^t v_G(\xi) \Psi^{(0)}_G(t-\xi) \,d\xi. $$ Applying the product rule for differentiation \begin{align*} \frac{\...
4
votes
0answers
88 views

Is a closed form possible for $\int\frac{\text{Li}_2(x)^2}{x}dx$?

Can $\,\displaystyle\int\frac{\text{Li}_2(x)^2}{x}dx\,$ be calculated by a sum/term of polylogarithm functions and the natural logarithm and polynomials (“closed form”) ? For the special case $\,\...
2
votes
3answers
47 views

How do I evaluate this indefinite integral?

I am currently working on the following problem: $\int x (2x+3)^{99}$ I have tried using u-substitution $(u = 2x+3)$ and integration by parts, but have not been able to make any progress that ...
4
votes
5answers
76 views

Evaluate the indefinite integral $\int \frac{dt}{(t^2-1)^2}$ of a rational function

I have this problem: $$\int \frac{dt}{(t^2-1)^2}$$ and I'm a bit unsure about how to proceed. I could use partial fractions: $$\frac{1}{(t+1)(t-1)(t+1)(t-1)}$$ $$\frac{1}{(t+1)^2(t-1)^2}$$ $$\...
1
vote
2answers
37 views

Hard indefinite integral with cube root

I'm stuck on evaluating this indefinite integral. $$\int\frac{dx}{x^2(1+x^3)^{\frac{2}{3}}}$$ I tried doing a u-substitution on the $1+x^3$ term inside the two thirds power but didn't get anywhere. ...
2
votes
0answers
91 views

Integral of $\int_0^b \cos(x)\cos(\frac a x)dx$

I've tried integration by parts. I can integrate both factors: $\int\cos(x)dx = \sin(x) + C_1$ $\int\cos(\frac a x) = a Si(\frac a x) + x \cos(\frac a x) + C_2$ However I'm stuck at this point. I ...