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

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1answer
40 views

Integral with quadratic square root inside trigonometric functions

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ ...
1
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0answers
29 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 ...
0
votes
0answers
9 views

Inverse Fourier transform of complex hyperbolic functions

I'm trying to solve a boundary condition problem and I got the solution in frequency regime: $$f(w)=\frac{\sinh(a|w|)}{b|w|\cosh(c|w|)-iw\sinh(c|w|)}$$ I'm wondering if there's any analytical form ...
1
vote
3answers
38 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
1answer
33 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. $C_1$ ...
0
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0answers
41 views

integration involving imaginary terms

How do we integrate forms of following type with imaginary terms involved? Can we get a closed form of it as result? ...
3
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0answers
45 views

Integration using exponent

What could be the techniques we need to use to solve this integration $\displaystyle \int\tan^2\theta\frac{\sin^2(\sec\theta\tan\theta)}{\sec^2\theta}d\theta \tag1$? How do I convert this in to a ...
0
votes
1answer
67 views

Integration with quadratic square root

What could be the techniques we need to use to solve this integration $\int\dfrac{s^2\sin^2\left(s\sqrt{ as^2+bs+c}\right)}{as^2+bs+c}ds$ ? Main issue here is the term inside $\sin^2()$. Very ...
1
vote
2answers
157 views

A Definite Integral I

Given the definite integral \begin{align} \int_{0}^{\pi} \frac{1+\cot^{2}(t)}{\cot^{2}(t)} \, \ln\left( \frac{1+2\tan^{2}(t)}{1+\tan^{2}(t)} \right) \, dt = -2\pi \end{align} then what is the general ...
0
votes
1answer
14 views

Proof regarding the primitives of periodic functions

Let $ f:R \to R $ be an integrable, periodic function. Prove that any primitive of such a function can be written as a sum of a periodic function and a function of the form $G(x)=ax$ where $a$ is a ...
4
votes
2answers
54 views

Closed form for $\int z^n\ln{(z)}\ln{(1-z)}\,\mathrm{d}z$?

Problem. Find an anti-derivative for the following indefinite integral, where $n$ is a non-negative integer: $$\int z^n\ln{\left(z\right)}\ln{\left(1-z\right)}\,\mathrm{d}z=~???$$ My attempt: ...
3
votes
3answers
86 views

Find $\int \dfrac{dt}{t-\sqrt{1-t^2}}$

Find $\int \dfrac{dt}{t-\sqrt{1-t^2}}$ MY APPROACH : Substitute $t = \sin x$ Multiply numerator and denominator by $\cos x+\sin x$ then rewrite everything in terms in $\sin2x$ and $\cos2x$, we ...
0
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1answer
32 views

how to prove the only difference between antidrivaties of a function is in their constants?

how to prove "If F is an antiderivative of f on an interval I , then the most general ...
0
votes
1answer
51 views

Integrals involving roots

I am bit stucked with an integration form while doing one of my proofs for a graphics application.Issue is I cant take out the terms from the trigonometric functions for a proper known integral ...
-1
votes
3answers
77 views

Find $\int (\arcsin x)^2\hspace{1mm}dx$ [closed]

Find $\int (\arcsin x)^2\hspace{1mm}dx$ $ $ How do we approach this problem
0
votes
1answer
30 views

Evaluating the indefinite integral $\int\frac{dx}{qx+c}$

Evaluate the indefinite integral (remember to use $\ln |u|$ where appropriate) $$\int\frac{dx}{qx+c}\qquad (q\neq 0) $$ I have no idea how to approach this. But here's what a have so far using ...
3
votes
2answers
103 views

Computing an indefinite integral

Let $\ P_n (x) = 1 + \frac{x}{1!} + \frac{x^2 }{2!} + \cdots + \frac{x^n }{n!} \ $ and $ I(x) = \int \frac{2n!\sin x + x^n }{e^x + \sin x + \cos x + P_n (x)}\, dx $ . (Where $\ n \to \infty \ $) ...
5
votes
2answers
88 views

How to Solve $ \int \frac{dx}{x^3-1} $

I am having quite a difficult time integrating $$ \int \frac{dx}{x^3-1} $$ My first approach was to apply a partial fraction decomposition $$ \int \frac{dx}{x^3-1} = \int \frac{dx}{(x-1)(x^2+x+1)} ...
6
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0answers
242 views
+300

Integration of product of functions(Special form)

Sir, I have been doing a proof related to one research topic. But after a long effort, I got ended up in a messy integration equation. Could you give me some suggestions to solve this equations? (Any ...
0
votes
2answers
67 views

Find $\int \sin^{-1}\sqrt{x}\hspace{1mm}dx$ [closed]

Find $\int \sin^{-1}\sqrt{x}\hspace{1mm}dx$ Can someone explain how to approach this problem
3
votes
2answers
48 views

Integrating $\sinh(x)\cosh(x)$

So I am very new to integration. I have to find the integral of $\sinh(x)\cosh(x)$ I have tried different ways: (i) let $u = \sinh(x)$, (ii) let $u= \cosh(x)$, and (iii) using the identity ...
2
votes
2answers
58 views

Integrating $x^3 \sqrt{x^2+1} $

I need to solve this integral $\int x^3 \sqrt{x^2+1} dx$ Someone could explain to me how? I have tried to use substitutions, i.e. $x^2+1$ and $\sqrt{x^2+1}$; but seems like its not correct. Someone ...
5
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4answers
92 views

Calculus Question: $\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx$

How to evaluate integral $$\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx?$$ I tried substitution $u^2=x^2-1$ and $u=\sqrt{x^2-1}+x$ but it turns out too complicated. Could anyone here help me to ...
1
vote
3answers
100 views

Find $ \int \frac{dx}{x\sqrt{1-x^4}}$

Find $\displaystyle \int \dfrac{dx}{x\sqrt{1-x^4}}$ I cannot figure out how start this problem, can anyone explain
1
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2answers
98 views

Find $\displaystyle \int \dfrac{\sin^2x}{1+\sin^2x}\hspace{1mm}\mathrm{d}x$

Find $$\int \dfrac{\sin^2x}{1+\sin^2x}\hspace{1mm}\mathrm{d}x$$ I cannot figure out how start this problem, can anyone explain
0
votes
5answers
68 views

Find $\int\sqrt{(x-2)/(x-1)}\,dx$

I am having difficulty with these type of problems. Can anyone explain how to approach the problems of the form $\int \sqrt{\dfrac{x-a}{x-b}}\hspace{1mm}dx$
3
votes
1answer
32 views

Integral evaluation with exponentials

I want to evaluate the integral $\int_0^T e^{-ax}e^{-bx^2} \, dx$. I found a direct solution: $$\int_{0}^{\infty} e^{-ax}e^{-bx^2} \, dx = \sqrt\frac{\pi}{b} \exp\left(\frac{a^2}{4b}\right) ...
0
votes
1answer
96 views

Integration with exponential

$$\int y\,e^{x^2}\,dy$$ I begin with $$\int e^{x^2}y\, dy$$ let $u=e^x$, $du=e^x\, dx$ how do I continue?
1
vote
1answer
38 views

Integrate and derivative

i'm not able to explain the following step: $\frac{1}{k+v(x)}=\frac{d^2 v}{dx^2}$ by integrating this equation: $(C-\frac{1}{k+v(x)})^{\frac{1}{2}}=\frac{dv}{dx}$ Please, if somebody can help i'll ...
0
votes
0answers
51 views

Evaluate $\int\tan x\, dx$

$-1 = 0$ by integration by parts of $\tan(x)$ The solutions to the above problem state that you can't cancel the integrals on each side because they both have an unknown constant attached to them. ...
2
votes
1answer
62 views

$\int \frac{1}{y-1} dy = \ln |y - 1|$?

I read the following in a book on differential equations $$\int \frac{1}{y-1} dy = \log |y - 1|$$ If I put $\int \frac{1}{y-1} dy$ into Wolfram Alpha it gives $\log (y - 1)$, i.e. the argument of ...
2
votes
3answers
65 views

Indefinite integral of a rational function: $\int\frac{6x+4}{x^2+4}\,dx$

Find $\displaystyle\int\frac{6x+4}{x^2+4}\,dx$ The question asks to find integral of the expression so I divided them into two parts: $$ \int\frac{6x}{x^2+4}\,dx $$ and $$\int\frac{4}{x^2+4}. $$ So, ...
2
votes
3answers
183 views

Simplifying expression and finding indefinite integral

(a) Simplify $$\Large \frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \quad.$$ (b) Hence find $$\Large \int\frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \mathrm{d}x$$ I tried to find a breakdown of the expression, but ...
4
votes
1answer
71 views

Indefinite integral typo in Gradshteĭn: reciprocal square-root of sixth degree polynomial

The indefinite integral below, quoted from Gradshteĭn's [Table of Integrals, Series, and Products][1], 7th ed., appears to contain at least two typos (highlighted in purple). $\mathbf{2.291.4}$ ...
1
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2answers
53 views

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal?

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal? I know both of these are the answer to $\int \csc x \space dx$, and I am able to work them out with calculus ...
1
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1answer
44 views

First and second derivatives of the function $f(x)=x\int_0^x e^{t^2}dt$

I haven't done calculus for a while so I need your help with these two exercises. I am not sure whether my solutions are correct so I'd really appreciate someone's feedback. $$ f(x)=x\int_0^x ...
2
votes
3answers
71 views

Struggling with $\int \frac{dy}{y\left(1 - \frac y2\right)}$

I know I need to use a partial fraction and suspect I will end up with 2 terms that end up as a natural log integral but I just can't work it out. $$\int \frac{dy}{y\left(1 - \frac y2\right)}$$ I ...
2
votes
2answers
61 views

Solve $\int (4x+2)\sqrt{x^2+x+1}\,dx$

Trying to solve this for a while now, so far I was able to come up without a proper answer. Problem : $\displaystyle \int (4x+2)\sqrt{x^2+x+1}\,dx$. I tried to take two common from $(4x+2)$ and ...
2
votes
3answers
58 views

How to Integrate $\int\frac{(x^2)}{\sqrt{7-x^2}}dx$.

I am trying to Integrate $\int\frac{(x^2)}{\sqrt{7-x^2}}dx$ and I have worked this problem a couple times and keep getting the same answer. So I will show my process and please point my errors out. ...
2
votes
1answer
46 views

Integrating $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I am trying to differentiate from my previous question, but I am having trouble in the finishing steps. I have the integral $\int\frac{\sqrt{16x^2-9}}{x}dx$. $$v=4x \hspace{15pt}dv=4dx$$ ...
1
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2answers
52 views

How to integrate $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I have the integer; $\int\frac{\sqrt{16x^2-9}}{x}dx$, and I am having trouble doing the trigonometric substitution. So for integrals in the from of $\sqrt{x^2-a^2}$ where $a$ is a constant is by ...
-1
votes
1answer
65 views

How to evaluate $\int \frac{1}{\cos^2 x (e^x + 1)} \,dx$?

The indefinite integral $$\int \frac{1}{\cos^2 x (e^x + 1)} dx$$ appears to be impossible to evaluate in closed form. Could you please suggest how I should evaluate this integral in definite form? ...
3
votes
2answers
90 views

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

I would like to ask for some help regarding the following indefinite integral, tried integration by parts and trigonometric substitution which both brought me to ...
1
vote
1answer
43 views

$\int_0^{2\pi} \sqrt{a^2 +b^2+2ab \cos\varphi}\,\mathrm{d}\varphi$

$$\int_0^{2\pi} \sqrt{a^2 +b^2+2ab \cos\varphi}\,\mathrm{d}\varphi$$ Where $a$ and $b$ are constants. I had to find the distance travelled by a point at a distance of $b$ from the centre of a rolling ...
4
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6answers
160 views

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

Evaluate $$\int\frac{\sqrt{1+x^2}}{x}\, dx$$ I tried substitution but failed miserably.
0
votes
1answer
11 views

indefinite trig substitution integral

I am not able to understand how to get the solution for an integral. Substituting something like $\frac{1}{12}tan(\theta)$ seems to be the right thing to do, but I can't figure it out from there. ...
1
vote
3answers
88 views

Evaluate $\int \frac{6x+4}{x^2+4}dx$

Find$$\displaystyle \int \dfrac{6x+4}{x^2+4}dx$$ I'm not really sure where to begin with this one - I know the answer will probably involve an $\arctan$, but I am unsure on how to use $\arctan$ in ...
1
vote
5answers
93 views

Evaluate $\int \frac{e^{-2x}}{e^{-2x}-3}dx$

$$\displaystyle\int \dfrac{e^{-2x}}{e^{-2x}-3}dx$$ I'm not sure how to integrate this. What's the first step? I thought it was the common result where the numerator is the derivative of the ...
2
votes
2answers
91 views

Calculate an integral with $\sin$? [closed]

How can I calculate the following integral? $$ \int \left[ \sin((x+x_0)T)\times\sin((x-x_0)T) \right]^{2n} \, \mathrm{d}x $$ where $x_0$ and $T$ are constants. Please help, thank you for your help ...
3
votes
1answer
58 views

Trig Substitution Problem - Integration

Suppose I want to integrate this (I chose an easy one): $$\int \frac {dx}{\sqrt{x^2-4}}$$ Method 1: (Trig Substitution) $x=2\sec(\theta)$ $$\int \frac {dx}{\sqrt{x^2-4}}=\int ...