Questions tagged [indefinite-integrals]

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

4
votes
1answer
94 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
75 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
35 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 ...
4
votes
1answer
52 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
44 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
53 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
36 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
49 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
68 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
42 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
85 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
49 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
93 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
69 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
27 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
66 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
64 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
46 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, ...
2
votes
2answers
139 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
51 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{\...
5
votes
0answers
104 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
48 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
79 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
44 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
100 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 ...
0
votes
3answers
65 views

Can you spot the mistake?

Let $$I=\displaystyle\int\sin^2x \ dx$$then $$I=\displaystyle\int(1-\cos^2x) \ dx=x+C-\displaystyle\int \cos^2x \ dx$$ Using the substitution $x=x+\frac{\pi}{2}$ we get $$I=x+C-\displaystyle\int\...
7
votes
2answers
126 views

Integrate $\frac{1}{x\,\log{x}}$ by parts [duplicate]

A naive indefinite integration of the function $\dfrac{1}{x\,\log{x}}$ can be performed as follows: Let $ \begin{eqnarray} I &=& \int\dfrac{dx}{x\,\log{x}}\\ \therefore I &=& \...
1
vote
4answers
89 views

How to integrate $\cos^3x$ by parts

I've converted $\cos^3(x)$ into $\cos^2(x)\cos(x)$ but still have not gotten the answer. The answer is $\dfrac{\sin(x)(3\cos^2x + 2\sin^2x)}{3}$ My answer was the same except I did not have a $3$ ...
2
votes
1answer
83 views

How to evaluate $\int\frac{1}{x^2\ln(x)} dx$ [closed]

I was trying to evaluate this integral, but I think it is a non-elementary function. $$I=\int\frac{1}{x^2\ln(x)} dx$$ Do you know any conditions where there is a closed form solution?
6
votes
1answer
68 views

Evaluate the indefinite integral

$ I = \int (x^2 + 2x)\cos(x) dx $ Integration by Parts, choose $u$: $$\begin{align} u &= \cos(x) \\ dv &= (x^2 + 2x)dx \\ du &= -\sin(x) \\ v &= \frac{1}{3}x^3 + x^2 \end{align} $$ ...
1
vote
1answer
37 views

Integral of $\frac{1}{\sqrt{(z-z')^2 + s^2}}$

I have a question about the signs of the antiderivative when one integrate $\frac{1}{\sqrt{(z-z')^2 + s^2}}$. According to Wolfram Alpha here and here: If one evaluates $\int \frac{1}{\sqrt{(z-z')^2 ...
3
votes
1answer
67 views

Evaluating an indefinite integral $\int \frac{dx}{\sqrt{x^3+2x+3}}$

Evaluate the following integral \begin{equation} J = \int \frac{dx}{\sqrt{x^3+2x+3}} \end{equation} I do not find suitable substitution to compute the above indefinite integral. Since $x^3+2x+3=(x+...
14
votes
1answer
529 views

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

I am trying to solve this integral but I can not figure what I do wrong. $$I=\int{\frac{1}{\sqrt{(2x^2+x+1)}}}dx$$ Here's how I go about it: I think that maybe it can be solved following the $$\...
2
votes
2answers
82 views

What is the inverse operation of a gradient?

I notice that the function $$f(x,y,x;a,b,c) = ke^{-a/x-b/y-c/z}$$ has partial derivatives $$\nabla f = \begin{bmatrix} \partial f / \partial x \\ \partial f / \partial y \\ \partial f / \partial ...
1
vote
2answers
32 views

Find the indefinite integral of $\int_{} \frac{x}{x^2+4}dx$

I am beginning to question whether the indefinite integral actually exists or I am doing something wrong with my u-substitution. Let $u = x^2 + 4, du = 2xdx,$ $$ \begin{align} \int_{} \frac{x}{x^2+4}...
0
votes
1answer
22 views

How to define the antiderivative of a function with singularities?

It seems like there are a few ways of describing the antiderivative of a partial real function with singularities. What are the different ways of nailing down more specifically what an antiderivative ...
0
votes
0answers
27 views

Cylinderial Shells:Volume of a bounded Region

I have to find the volume of the bounded region by the following functions: $$y=\frac{1}{x^3}$$ $$y=0$$ $$x=1$$ $$x=2$$ $$\mathbf{Revolving\ around\ the \ axis:}$$ $$x=-1$$ The method ...
2
votes
2answers
128 views

What would be the integral of the zeta function or $ \sum\limits_{n=1}^{\infty} \frac {1}{n^x} $?

The zeta function is defined as: $$ \zeta (x) = \sum\limits_{n=1}^{\infty} \frac {1}{n^x} $$ Does an integral of this function exist? If it does then what would it be? More information about zeta ...
1
vote
2answers
67 views

Integral of $\ln(\arcsin(x))$

I tried a u-substitution of $u=\arcsin(x)$ but that resulted in a bunch of bad square roots. I also looked it up on Wolfram Alpha and it has the $\text{Si}$ function but I don't really know what that ...
0
votes
2answers
62 views

Integrate $x^2 \sqrt[]{3+5x^2}dx$ (preferably) using substitution

Today we went over solving integrals with tables. My task is to integrate the following: $\int x^2 \sqrt[]{3+5x^2}dx$ In the back of the book, I am provided with over $100$ integrals. I believe ...
0
votes
3answers
43 views

Find the indefinite integral $\int_{} \frac{t^3 dt}{\sqrt{1+t^2}}$ [duplicate]

Find the indefinite integral $\int_{} \frac{t^3 dt}{\sqrt{1+t^2}}$ 1) Choose a u I decide to let $$u = 1 + t^2, du = 2t$$ 2) Substitute u value into integral $$\int_{} \frac{t^3}{\sqrt u} du$$...
0
votes
4answers
24 views

Find the indefinite integral via substitution rule

Find the indefinite integral $\int_{} (\cos^3x)(\sin x)\mathrm dx$ Here is my work. 1) Pick the $u, v$ values: $$u = \cos x, \mathrm du = -\sin x$$ $$v = x, \mathrm dv = 1$$ 2) Substitute $u, ...
0
votes
0answers
22 views

How to find asymptotics of indefinite integrals

Often, an indefinite integral cannot be expressed exactly. It would be useful if it were possible to at least prove something about asymptotics of the indefinite integral. I wasn't able to find much ...
1
vote
2answers
84 views

How to reduce an integral with square root of cubic function into an elliptic integral

I need to calculate the following ntegral: $$\int \frac{t }{\sqrt{2 t^3 - 3 t^2 + 6 C}} dt$$ where $C$ is a constant to be determined later, so I cannot look for roots of the polynomial in the ...
4
votes
2answers
102 views

How to find $\int \sum_{n=0}^\infty \frac {(\ln x)^n}{x^n.n!}dx $?

I know how to find the integral of a normal power series like the power series of $ \sin (x) $, but have no idea how to integrate the $( \ln x )^n $ term inside the summation.
0
votes
2answers
86 views

Integral $\int \exp\left(-\frac{z^2}{2}\right) \frac{z^4+2z^2-1}{(z^2+1)^2}dz$

Integrate $$\int \exp\left(-\frac{z^2}{2}\right) \frac{z^4+2z^2-1}{(z^2+1)^2}dz$$ This function has exponential factor and rational factor. I know already that the solution is $$-\frac{z\exp\left(...
1
vote
1answer
188 views

How would we find $ \int \sqrt[x]{x}$ or $\int \left( x^{x^{x^{.^{… \infty}}}} \right) dx$?

I know that the antiderivate of $$x^{\frac{1}{x}}$$ cannot be expressed in the form of elementary functions, but even so how will we express it in the form of a power series ? Inspiration of the ...
-4
votes
1answer
44 views

Integral of $ \int \left( 1 + \frac {\log (x)}{x} + O \left( \left( \frac {1}{x} \right)^2 \right) \right) dx $?

How would we integrate a function with a Big O notation like this: $$ \int \left( 1 + \frac {\log (x)}{x} + O \left( \left( \frac {1}{x} \right)^2 \right) \right) dx $$
6
votes
3answers
178 views

What is the series expansion of $ \sqrt[x] x $?

What is the series expansion of $ \sqrt[x] x $ ? I want to find it because I want to find $ \int \sqrt[x] x dx$ but believe that the integral cannot be expressed in the form of elementary functions.