Tagged Questions

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

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2
votes
1answer
64 views

how to integrate $\mathrm{arcsin}\left(x^{15}\right)$?

Integral by parts: $$ I = x\sin^{-1}\left(x^{15}\right) - \int\frac{15x^{15}}{\sqrt{1-x^{30}}}dx $$ then what? The answer by wolfram gives an answer contains hypergeometric ${}_2F_1$ function,because ...
1
vote
3answers
301 views

Integral of $(2-x)/(x-1)$

So I tried doing this: I have $$\int \frac{2-x}{x-1} \mathrm{d} x$$ I used the substitution $u = x-1$, thus $x= u+1$ and $ \mathrm du = \mathrm dx$. So then our integral becomes $$\int ...
0
votes
0answers
87 views

Nasty integration?

So I am trying to solve the following integral and apparently its not integrable or I might be wrong. Not even computer software can integrate. Can anyone tell me if this is integrable or not? The ...
2
votes
1answer
48 views

Why am I obtaining an imaginary part for my integration

I try to solve an integration as follows, $$\int \frac{sy^{-1}}{(1+sy^{-1})} \text{exp}(-\sqrt{y})dy$$ as you can see its pretty complicated, and I get an answer like the following using Wolfram ...
3
votes
0answers
59 views

Solving double integrals numerically?

I have written this in way to make it as much as possible non-confusing. I will start describing my problem and I will walk you through my question, I have a double integration which I am trying to ...
2
votes
0answers
22 views

integral substitution that use the derivative

I tried to solve the equation $$\frac{dy}{dx}=\frac{k}{\sin{x}\sqrt{\sin^2{x}-k^2}}$$ and was suggested the substitution $$ \tan{u}=\frac{dy}{dx}\sin{x} $$ after some algebra (I'll add it if ...
3
votes
3answers
45 views

Evaluating indefinite integral using a trigonometric substitution

I have this integral: $$\int\frac{x^3}{\left(\sqrt{4x^2+9}\right)^3}\,dx$$ I tried to solve it with a trigonometric substitituon but I can't get any result. I would appreciate if somebody could help ...
4
votes
1answer
50 views

Calculating indefinite integral?

I want to calculate $$I_n = \int \frac{d\theta}{\sin^n(c\theta)\cdot \cos(c\theta)}. $$ The answer is $$-\frac{1}{c(n-1)\sin^{n-1}(c\theta)}+ ...
4
votes
0answers
90 views

How to compute or simplify this nasty integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
2
votes
1answer
25 views

Integration of $x^a$ and Summation of first $n$ $a$th powers

I'm learning some discrete mathematics. I already knew a little (very little) calculus, and I noticed something. I think it's just a coincidence, so I'm sorry if this is a bad question. There are some ...
0
votes
0answers
27 views

Help in evaluating the integral at the given limits

Hi guys I am hoping to get some help here. The indefinite integral below gives the following result. $$\int \left(\frac{0.0016 \left(1-\exp \left(-0.0112 v^{0.25}\right)\right)}{v^{0.5}}+\frac{0.0036 ...
1
vote
1answer
51 views

Help in computing this integration?

Any thoughts or hints on solving the following integral $$ \int_{y}^{+\infty} \frac{ e^{-\sqrt{v}} }{1+ s v^{-1} } dv $$ and where $$s= \frac{2}{x^{-1}+y^{-1}}$$ The result should be a function ...
1
vote
1answer
29 views

Why would the derivate of u - 2 = x be du = xdx?

I am going through an integration requiring a u-substitution on a practice midterm, and the professor posted a solution which used: u = x - 2 u - 2 = x du = xdx I am confused how du could be ...
-4
votes
1answer
38 views

integrate by parts: $\int \cosh^2(x)dx$ please show solution step by step [closed]

Integrate by parts: $$\int \cosh^2(x)dx$$ Please show the solution step by step. I actually somehow found my self in a loop solving the integral: = cosh(x) sinh(x) - int (sinh(x) (-sinh(x)) (x) = ...
3
votes
2answers
37 views

Find $\int{\frac{1}{\left(1+\ln x\right)^2}\;dx}$

How would you integrate a function almost entirely in logarithmic form, such as:$$\int{\frac{1}{\left(1+\ln x\right)^2}\;dx}$$ I have tried various substitutions and considered integrating by parts, ...
1
vote
1answer
16 views

how to evaluate $\int \left(\hat{V}\times \frac{d^2\hat{V}}{dt^2}\right)dt$?

if $\hat{V}\left(t\right)$ is a vector function of $t$, find the indefinite integral $\int \left(\hat{V}\times \frac{d^2\hat{V}}{dt^2}\right)dt$ To solve thi first i find for the integrand with ...
3
votes
1answer
54 views

Is this an acceptable way to integrate?

I am supposed to find: $$ \int \sec(1-x)\tan(1-x) dx $$ I then set $ u = \sec(1-x) $ $$ du = -\tan(1-x)\sec(1-x)\ dx $$ therefore $$ \frac{-du}{\sec(1-x)} = \tan(1-x)\ dx$$ Which when applied gives ...
11
votes
2answers
125 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot ...
0
votes
1answer
40 views

How can the integral of $|\sin(x)|$ be $-\cos(x)\text{sgn}(\sin(x))$?

Wolfram|Alpha tells me that $\int|\sin(x)| = -\cos(x)\text{sgn}(\sin(x))$ (which happens to also be its derivative), but I don't understand how this is possible, because the resulting function jumps ...
7
votes
4answers
137 views

Integrate $\int\frac{dx}{(x^2+1)\sqrt{x^2+2}}$

I would like some guidance regarding the following integral: $$\int\frac{dx}{(x^2+1)\sqrt{x^2+2}}$$ EDIT: The upper problem was derived from the following integral ...
2
votes
1answer
47 views

Integrate $\sin^n{x}$

How do you integrate: $\int(\sin^n{x}) dx$ The link to WolframAlpha : (Integration Answer) No definite limits... What is that hypergeometric function in that answer. Please help! Thanks
8
votes
4answers
134 views

Evaluate $\int(x^{91}+x^{327})\cos(x)\mathrm{d}x \quad .$

Evaluate $$\int\left(x^{91}+x^{327}\right)\cos(x)\mathrm{d}x \quad .$$ It's my first time to face integration like that. I just need a clue to start because I tried, but it's not working Thanks in ...
1
vote
3answers
49 views

Integrating $\;\int x^3\sqrt{x^2 + 2}\,dx$

Integrate the following: $$\int x^3\sqrt{x^2 + 2}\,dx$$ I understand how to do basic integration by parts but I don't know what to do with $\sqrt{x^2+2}$. Do I divide the $\sqrt{x^2+2}$ by 2 ...
1
vote
2answers
72 views

Integral $\int \frac{\operatorname d \! x}{\sinh^4 x}$

How to evaluate: $$\int \dfrac{\operatorname d \! x}{\sinh^4 x}$$ I tried to split it in $\int \frac{1}{\sinh^2x}\frac{1}{\sinh^2x}$ and then integrate by parts, but it's seems to complicate the ...
1
vote
1answer
45 views

Integral of a function which yields a hyper-geometric function

Note that $n$ is an arbitrary constant. $$ \int(\sin^n(x))dx $$ I start by using the obvious integrating by parts and get: $$ \frac{d}{dx}[x\sin^n(x)] = \sin^n(x) + nx\sin^{n-1}(x)\cos(x) $$ $$ ...
0
votes
2answers
38 views

Volume of a horizontal cylinder using height of liquid

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank. We need to determine the amount of liquid ...
1
vote
1answer
22 views

Total Mass of a Spherical Object

Consider a spherical galaxy with volumetric mass density, at a distance $s$ from the center, is given by $$ \rho = \frac{k}{1+s^3} $$ where $k$ is a constant. Let $k = 25$. Determine the total mass ...
0
votes
3answers
69 views

Integral using height to find volume

How do you find the volume of a "pit" which is circular in horizontal cross-section, and parabolic in vertical cross-section using height by "sticking". "Sticking" is when we insert a dipstick through ...
1
vote
2answers
57 views

find $\displaystyle \int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$

find $\displaystyle\int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$ If I do Integration by parts, I end up with $\displaystyle\int e^{-2x-x^2}\hspace{1mm}dx$ Which I believe cannot be ...
0
votes
1answer
38 views

Solving an Integral - $ \int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt $

How do we solve $ \int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt \tag 1 $ to a finite form? $k,a,b,c$ are constants $at^2+bt+c$ does not guarantee equal roots always
1
vote
1answer
39 views

Evaluating an indefinite integral with an inverse trigonometric function

I'm really stumped on a homework problem asking me to evaluate $\int \frac{ln\ 6x\ sin^{-1}(ln6x)}{x}dx$, and after a few hours of trying different approaches I'd definitely be appreciative for a bump ...
5
votes
2answers
95 views

About Integration

How to calculate the following integral $$ \int \frac{\tanh(\sqrt{1+z^2})}{\sqrt{1+z^2}}dz $$ Is there any ways to calculate those integral in analytic? (Is $[0,\infty]$, case the integral is ...
1
vote
1answer
38 views

Indefinite integral with trig components

The following integral has me stumped. Any help on how to go about solving it would be great. $\int\frac{\cos\theta}{\sin2\theta - 1}d\theta$
4
votes
2answers
77 views

Evaluation of $\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $

How do we find $$\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $$ NB: It is not mandatory that $ax^2+bx+c$ has only a single root
2
votes
2answers
61 views

Integration techniques for $\int x^3\sin x^2\,dx$

I've learned a couple of methods of integrating, but I'm still not sure when to use which one. Example problem is \begin{align} \int x^3\sin x^2\,dx \end{align} I tried using a method where I set ...
5
votes
2answers
96 views

Evaluation of $\int \frac{x\sin(\sin x)}{x+5} \ dx$

How do we find $$\int \frac{x\sin(\sin x)}{x+5} \ dx\ ,$$ is there any way to take that $\sin x$ out from parent $\sin(\cdot)$ ?
0
votes
0answers
26 views

Integration by Parts on Square Matrix

Note : " ' " implies derivative w.r.t s Given Data in the Problem We have given matrix functions $R(s)_{3\times 3}$ and $K(s)_{3 \times 3}$. It has following relationships $R(s)^{'}_{3\times ...
3
votes
4answers
96 views

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$ $\bf{My\; Try}::$ Let $\displaystyle I = \displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$, Now let $\displaystyle \tan ...
0
votes
1answer
54 views

Indefinite integral $\int t \cdot \cos^3(t^2)dt$

I am having trouble integrating $$\int t \cdot \cos^3(t^2)dt$$ Progress I have made $u=t^2$ which makes the problem $1/2 \int \cos^3(u) du$. After writing that out I subsituted $v=\sin(u)$ ...
2
votes
1answer
60 views

Integrate: $ \int \frac{\mathrm{d}x}{\ln(x)} $

I am having quite a bit of difficulty integrating, $$ \int \frac{\mathrm{d}x}{\ln x } $$ I believe a u-substitution will not work since if $ u = \ln(x) $ then $ \mathrm{d}u = \frac{\mathrm{d}x}{x} $ ...
1
vote
0answers
33 views

Integration by Parts in matrices

Given Data in the question We have a given equation based on matrices as follows $\frac{\mathrm{d} R(s)_{3\times3}}{\mathrm{d} s}=R(s)_{3\times3}K(s)_{3\times3} \tag 1$ $\frac{\mathrm{d} ...
-1
votes
0answers
13 views

Integration over ordered random variables

I have a joint distribution function over random jointly distributed random variables $(X,Y)$ denoted by $f_{X,Y}(x,y)$. Assuming without loss of generality that $$X<Y$$ I would like to find ...
5
votes
1answer
31 views

Integral versus hypergeometric series: how to solve this?

How can I resolve the following indefinite integral using hypergeometric series? $$ \int (x^3 + 1)^\frac{1}{3} \,dx $$ Wolfram Alpha indicates that the series of Appell are used, but how to get to ...
0
votes
1answer
41 views

Compute integral containing a matrix

Let $\mathbf{H}= \begin{pmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{pmatrix}$ and $P(\mathbf{H}$) the joint probability distribution of $\mathbf{H}$ given by: $e^{-(a+ ...
0
votes
4answers
50 views

Definite Integrations problems [closed]

If $f(x)= x^2 e^{x^2}$ then show that $f'(x)= 2xe^{x^2} + 2x^3 e^{x^2}$ and use this result to evaluate $$\int x^3 e^{x^2} \, dx$$ How can I use the result to evaluate the integral?
5
votes
2answers
199 views

Simplest way to integrate this trigonometric integral:

$$\int \frac{1}{1+\tan x}dx,$$ A substitution like $t = \tan x, \;dt = (1+t^2)dx$ etc. immediately comes to mind, but I find this method a bit lengthy with the partial fractions. Is there a more ...
0
votes
0answers
22 views

Question from integral with using fourier's integral

Please explain me how to compute this integral: $$ \int_0^\infty \dfrac{\cos(\omega x)+\omega \sin(\omega x)}{1+\omega^2}d\omega$$
0
votes
2answers
69 views

Calculate $\int(1-\sin x)^2\cos x\,dx$ [closed]

How to calculate the following integral? Calculate $\displaystyle\int(1-\sin x)^2\cos x\,dx$.
3
votes
1answer
77 views

How to evaluate the following integral? $\int\frac1{1+\sqrt{\tan x}}\mathrm dx.$

Evaluate the following integral: $$\int\dfrac1{1+\sqrt{\tan x}}\mathrm dx.$$ I know this question has a solution, but I haven't the slightest idea how to do it.
2
votes
3answers
60 views

Integration question: $\int \frac{\mathrm{d}x}{\sqrt{3 x} (3 x+1)}$

I am missing one piece of how to integrate the following: $\int \frac{\mathrm{d}x}{\sqrt{3 x} (3 x+1)}$ I found a solution to a similar problem which I entirely understand: I can usually use ...