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
53 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
122 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
39 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
126 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
45 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
131 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
46 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
68 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
44 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
35 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
19 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
66 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
36 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
93 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
75 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
57 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 $\sin(\cdot)$ ?
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 $v=\sin(u)$ ...
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
12 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 ...
0
votes
1answer
39 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
48 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
196 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
66 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
76 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 ...
0
votes
1answer
42 views

$\int e^{x^2}(y-1) \,dx$ [closed]

What is $\int e^{x^2}(y-1) \,dx$ ? I could not find the answer.
1
vote
0answers
17 views

What technique is the last step of this integration?

So, in my book, Circuit Analysis, Second Edition by Cunningham and Stuller the give the following integration for the the energy absorbed by a capacitor in circuit. Remember $p=vi$ (power absorbed = ...
0
votes
2answers
63 views

Integrate $\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$ [closed]

Evaluate$$\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$$ Solution through either contour integral method or indefinite integral method please!
3
votes
4answers
103 views

Integrating $\int \dfrac {dx}{\sqrt{4x^{2}+1}}$

$\int \dfrac {dx}{\sqrt{4x^{2}+1}}$ I've been up to this one for quite a while already, and have tried several ways to integrate it, using substituion, with trigonometric as well as hyperbolic ...
2
votes
1answer
53 views

Integration of a function containing inverse trigonometric functions

Q. $$\int \sin\left\{2\tan ^{-1}\left(\sqrt{\frac{3-x}{3+x}}\right)\right\}dx$$ $\implies$ $$\int \sin\left\{\sin ...
3
votes
2answers
49 views

Are there examples of when the ILATE mnemonic for choosing factors when integrating by parts fails?

Is it possible in some cases that using the ILATE rule does not yield an explicit antiderivative but making another choice does yields one? If so, please give examples.
5
votes
1answer
83 views

Integrals $\int \frac{1}{\operatorname{arctanh}(x)} \, dx$ and $\int \frac{1}{\operatorname{arccoth}(x)} \, dx$

Do we know anything about this integrals? $$ \begin{align} I_1(x) = \int \frac{1}{\operatorname{artanh}(x)} \, dx \\ I_2(x) = \int \frac{1}{\operatorname{arcoth}(x)} \, dx \end{align}$$ Similar ...
2
votes
2answers
25 views

Multiple answer for integration of a function?

Q. $\int \left(\frac{sin2x}{sin^4x+cos^4x}\right)\:dx$ My method: $$\int \:\left(\frac{sin2x}{sin^4x+cos^4x}\right)\:dx=\int ...
2
votes
3answers
104 views

How to find the antiderivative of this function?

I want to integrate this function: $$\int\dfrac{x^2}{e^x-1}dx$$ I used integration by parts formula to integrate it. However I have reached somewhere where I got something like this: ...
0
votes
1answer
21 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
74
votes
9answers
4k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
2
votes
3answers
88 views

Calculus 2 Integral of$ \frac{1}{\sqrt{x+1} +\sqrt x}$

How would you find $$\int\frac{1}{\sqrt{x+1} + \sqrt x} dx$$ I used $u$-substitution and got this far: $u = \sqrt{x+1}$ which means $(u^2)-1 = x$ $du = 1/(2\sqrt{x-1}) dx = 1/2u dx$ which means ...
10
votes
1answer
122 views

Integral of $\sqrt{x^3 + 8}$?

I have issues solving the following integral: $$\int\sqrt{x^3+8}~dx$$ I tried substitution and integration by parts, but with no use. I'm guessing I have to use some trigonometric substitution. ...
0
votes
4answers
71 views

What is the most efficient way to integrate $(x-3)\sqrt{x^2+3x-18}$?

I can do the problem, but it is becoming so big,that I do not feel to do it anymore. Can anyone give the shortest method for this problem? $$\int (x-3)\sqrt{x^2+3x-18}\,dx $$
1
vote
2answers
65 views

Indefenite Integral requiring substitution

Can someone please help me find a useful substitution for the following integral: $$\int \frac{1}{\sqrt{x}(1+\sqrt{x})^2}dx$$ I tried letting $ u = \sqrt{x} $ But I couldn't proceed. Please help.
1
vote
0answers
37 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
1
vote
3answers
42 views

finding an indefinite integral of a fraction

(a) Show that $\frac{4-3x}{(x+2)(x^2+1)}$ can be written in the form ${\frac{A}{x+2} + \frac{1-Bx}{x^2+1}}$ and find the constants $A$ and $B$. (b) Hence find ...
1
vote
2answers
60 views

Integral involving exponents

How do we integrate $\int e^{C_1\frac{u^2+1}{u^2-1}} \ du\tag 1$ I could not find a proper substitution to convert it to a normal available form so that I can get a closed form of integration. $C_1$ ...