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

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2
votes
2answers
54 views

Integrate $ \int \frac { e^{\arctan(x)}}{{(1+x^2)}^{\frac{3}{2}}} \ dx $

$y=arctanx$ $tany=x$ \begin{align} \int \frac { e^{\Large\arctan(x)}}{{(1+x^2)}^{\Large\frac{3}{2}}} \ dx&=\int \frac {e^{\Large\arctan(\tan y)}}{{(1+\tan^2y)}^{\Large\frac{3}{2}}}dy\\ &=\int ...
0
votes
0answers
24 views

Hints to find analytical solution to integral

I have to evaluate the expression $$f(|\vec{c}|) = \int_0^\infty \int_0^{2\pi} (z(\vec{a})+z(\vec{a}+\vec{c})) \frac{(1-\cos(\theta_{\vec{a}+\vec{c}} - ...
0
votes
1answer
42 views

Integral $\int\frac{dx}{(x^3-1)^2}$

Please help. I do not know what to do. You can just show the direction where to go and I continue. Here it is: $$\int\frac{dx}{(x^3-1)^2}$$
1
vote
4answers
69 views

Help evaluating $ \int \sqrt{{x}^{2} + 3} \; dx $

Can you help me evaluating the following indefinite integral? $$ \int \sqrt{{x}^{2} + 3} \; dx $$ Please, don't give a full solution, just some hint on which method to use...
4
votes
2answers
59 views

Integral of $\big((1+\cos(x))\sin(x)\big)^2$

What is $$\int \big((1+\cos(x))\sin(x)\big)^2dx$$ ?
0
votes
1answer
55 views

A indefinite integral $\int 2(1+x^4)^\frac{1}{3}dx $

how can i solve this indefinite integral: $$\int 2(1+x^4)^\frac{1}{3}dx $$ I need a solution!
4
votes
2answers
47 views

Is there a notion of indefinite Lebesgue integral?

When I started studying integration rigorously via the Riemann and Lebesgue integrals, one thing that struck me is that we loose completely the concept of indefinite integrals. Integrals of functions ...
0
votes
1answer
88 views

Computing $\displaystyle \int{ \frac{\sqrt[3]{x}+2\sqrt[4]{x}}{\sqrt{x}(\sqrt{x}+\sqrt[3]{x})^2}}\,dx$

$$\int{\sqrt[3]{x}+2\sqrt[4]{x}\over \sqrt{x}(\sqrt{x}+\sqrt[3]{x})^2}dx$$ I think i need to make a substitution but am having difficulties because of the different roots. $$\int \left(\sqrt{x \over ...
0
votes
1answer
39 views

Is there another way than linearization?

$$I= \int {\sin^mx \cos^nx }dx$$ I need a Hint on doing this integral a Successive Partial Integration but it seems that the problem shows up when $ m = 2k $ and $ n = 2p$ where $p,m \in \mathbb{N}$. ...
0
votes
1answer
30 views

Volume of a region [closed]

I could not figure out how I can implement this question with using integration by parts. Any help please. Find the volume of the solid generated by revolving the region bounded above $y=2\cos x$ and ...
2
votes
4answers
98 views

Integration by parts, What I should do next

I'm integrating a function by parts but I get stuck in a part. Since here, What should I do next?
0
votes
2answers
36 views

Indefinite Integral:Can anyone kindly give an appropriate technique?

Indefinite Integral:I tried this integral by substitution,but cannot find any appropriate derivative. I tried it by product rule but ended in a complex form. Can anyone kindly give an appropriate ...
1
vote
5answers
57 views

Evaluate $\int x^3(x^2+7)\ dx$

I'm trying to find the indefinite integral of $$\int x^3(x^2+7)\ dx$$ and I've seem to have forgotten how to do it in this case. So if anyone can refresh my memory, I'd appreciate it.
5
votes
1answer
31 views

Integration of $\int \frac{\arcsin{e^x}}{e^x}dx$

I've got a problem with this integral: $$\int \frac{\arcsin{e^x}}{e^x}dx$$ I got such a result: $$\int\frac{\arcsin{e^x}}{e^x}dx=-\frac{\arcsin{e^x}}{e^x}-\ln|\sqrt{e^{-2x}-1}+e^{-x}|+C$$ but the ...
2
votes
4answers
81 views

Integral of $\int\sin^{3}xdx$

In evaluating integral $\int\sin^{3}x dx $ I am pretty sure we need to use substitution $e^x=t$, but can't go next step.
4
votes
1answer
26 views

Does picking $C=0$ as a constant of integration result in a nominated anti-derivative?

Introductory calculus students are often introduced to the "indefinite integral" or anti-derivative before actually doing integrals because it makes the FTC seem natural (by some rather sketchy ...
1
vote
0answers
25 views

Is it possible to derive a closed-form analytical expression when integrating over a triangular area of a bivariate Archimedean copula PDF?

Let's use, for example, one of the simpler Archimedean copulas - the Clayton copula with $\theta>0$. What I want to calculate is the probability associated with, say, a triangular region of ...
1
vote
2answers
58 views

Evaluating: $ \int\sqrt{\tanh(\ln(\sqrt{x}))} dx$ ; $ \int \ln\left(\sqrt{\tanh(\ln(\sqrt{x}))}\right) dx$

I don't have much experience with hyperbolic trig funnctions... So I don't know how to start solving this. How do I evaluate the following integrals? Any advice, hint or well-thought solution will be ...
2
votes
3answers
77 views

Integration of $\int\log(\sqrt{1-x}+\sqrt{1+x}) \, dx$

Integration of $$\int\log\left(\sqrt{1-x}+\sqrt{1+x}\right) \, dx$$ Please help to go through this problem as i have started with putting $x$= $cos2y$.
3
votes
2answers
66 views

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

How to tackle with this problem $$\int \frac {1}{1+x \sqrt{1-x^2}} dx$$ I have put $x$ = $\sin y$. But couldn't reach at the final result. Please help....
1
vote
2answers
22 views

Interpreting the integral identity $\int c f(x) \,dx = c \int f(x) \,dx$ for $c = 0$

Let me begin my question with these 2 facts: The function $F(x)=3$ is an antiderivative of the function $f(x)=0$ since $F'(x)=f(x)$ Constant Multiple Rule: $\int cf(x)dx=c\int f(x)dx, \forall c \in ...
2
votes
2answers
80 views

Evaluate the following indefinite integral

Evaluate the integral : $$\int\frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}\,dx$$ I tried through putting $x=\tan \theta$ as well as $x=\tan^2\theta$ .but I am unable to remove the square root. I also tride by ...
4
votes
3answers
69 views

stuck with (last) partial integration step for $\int x^2 e^{2x} \, dx$

I am stuck with this integral in the last step of partial integration: \begin{align} \int x^2 e^{2x}\,dx & = \frac{1}{2}e^{2x}x^2-\int \frac{1}{2}e^{2x} 2x \,dx \\[6pt] & = ...
3
votes
5answers
78 views

How to calculate $\int \sqrt{x^2+6}\,dx$?

How to calculate $\int \sqrt{x^2+6}\,dx$, by using Euler substitution and with to use of the formula : $\int u\,dv = vu - \int v\,du $. note: what I mean by Euler substitution: is when we have a ...
2
votes
3answers
73 views

Integral of $\int \frac{\sin x (2 \cos x - \sin x)}{2\sin x + \cos x} dx$

Integrate the following integral: $$\int \frac{\sin x (2 \cos x - \sin x)}{2\sin x + \cos x} dx$$ I have tried it by using by parts by considering the $\sin x$ as first function. Again in the ...
0
votes
1answer
49 views

Integral of $\int \frac{x \sin{x} \cos{x}}{(a^2 \sin^2 x + b^2 \cos^2 x)^2} dx $

Please help me to find the integral of $$\int \frac{x \sin{x} \cos{x}}{(a^2 \sin^2 x + b^2 \cos^2 x)^2} dx $$ There is a problem in having x in the numerator. Please guide me . Thanks in advance.
1
vote
5answers
57 views

How to solve integration of a product of an exponential and a trigonometric function?

Preparing for the exam I bumped into this integral and I just can't get hold on it. It's an integration of a product of an exponential and a trigonometric function. It's going in an endless loop for ...
0
votes
1answer
59 views

Integrate $\int \frac{\sin(3x)}{\cos x}~dx$

An answer is $\ln|\cos x| - \cos2x$. I'm trying to get the answer but I'm getting something different $$\int \frac{\sin(3x)}{\cos x}~dx = \int\frac{3\sin x - 4\sin^3x}{\cos x}~dx=3 \int \tan x ~dx ...
5
votes
3answers
130 views

Evaluate $\int \frac{1}{x^3+3x+1}dx$

Evaluate $$\int \frac{1}{x^3+3x+1}dx$$ I tried to evaluate it but I couldn't do .
-2
votes
4answers
83 views

How to calculate : $\int \frac{e^{3x} +1}{e^x + 1} dx$ [closed]

How to calculate : $$\int \frac{e^{3x} +1}{e^x + 1} dx$$ I'm trying to calculate it without substitution, just using normal formulas for integration, how can I do this?
2
votes
2answers
80 views

Closed form for $\int \frac{1}{a^x + x^a} dx$

Is there a closed form for the following integral? I can't seem to find it online, nor do I have any idea how to approach this problem..... $$\int\frac{1}{a^x + x^a}dx$$ where $a \ne 0 ; a\in R $
-1
votes
2answers
33 views

Integration of logarithm

$\int \ln(\ln \sqrt{x})^{\ln (x)}dx$ how should I integrate this? I think it can't be integrated. I don't know.
2
votes
1answer
40 views

Find $\int \left(e^{\frac{x}{2}}-\frac{5}{x^2}\right)dx$ by substituion

$\int \left(e^{\frac{x}{2}}-\frac{5}{x^2}\right)dx$ I am supposed to use u substitution, but I'm not sure which variable to choose as u. Any help is appreciated, thanks!
2
votes
3answers
84 views

Theoretical or intuitive understanding of the integral $ \int_\ \frac{dx}{x} = \ln|x|$?

I see that it works, but I'm not sure I get why. Both the relationship between (1/x) and the ln function itself, which seems bizzar, and the absolute-value part (I get how you need it for negative ...
1
vote
1answer
34 views

Why $\int \frac{\ln(x)^n}{(x-s)(x-s_0)} dx \propto \ln(x)^{n+1}$

I am currently trying to find an argument for the fact $\int \frac{\ln(x)^n}{(x-s)(x-s_0)} dx \propto \ln(x)^{n+1}$ This seems to be approximatively the case from looking at the first few powers of ...
0
votes
1answer
39 views

Simplifying an expression involving integrals: $ \xi= (e^\Phi)(\Omega^2)(\Psi^{-1})$

Simplify the following expression: $$\Large \xi= (e^\Phi)(\Omega^2)(\Psi^{-1})$$ Where: $$ \large ...
2
votes
2answers
98 views

How can I evaluate: $\large \int \psi^{x^{\phi/\psi}} dx$

I'm having trouble evaluating the following integral: $$\large \int \psi^{x^{\frac{\phi}{\psi}}} dx$$ Question: How can I evaluate this integral? (Note: $\phi$ and $\psi$ are constants) Note: I ...
1
vote
1answer
67 views

How to evaluate $\int \sqrt{\sin^{-1}(\sqrt{\phi})} d\phi $?

How do I go about solving the below integral? $$I_1=\int \sqrt{\sin^{-1}(\sqrt{\phi})} d\phi $$ Background: I came across the simpler version of this, which required me to evaluate: ...
1
vote
4answers
85 views

How to integrate $\int \frac{dx}{x^2 \sqrt{x-1}}$?

I need to integrate$$\int \dfrac{dx}{x^2 \sqrt{x-1}}.$$ I've tried everything from substitutions ($\sqrt{x-1}=u$) to integration by parts but I cannot get anywhere. Please help.
3
votes
2answers
53 views

Integral $\int \frac{x^2}{\sqrt[] {3-x^3} } \operatorname d \! x$

$$\int \frac{x^2}{\sqrt[] {3-x^3} } \operatorname d \! x$$ I have the above indefinite integral problem with substitution and I wanted to see if I could get a double check from the masses just in ...
3
votes
1answer
40 views

indefinite integral substitution trickery

I want to find the indefinite integral using substitution on the above. I needed a few sets of eyes and also maybe the appropriate dunce cap if need be. I am wondering if I am on the right track, ...
0
votes
1answer
35 views

Help me solve the indefinite integral

So, I ran across this in my workbook and have no idea where to start. I can't recognize any basic form or any substitution which I can make. $$I=\int\frac{dx}{\sqrt[n]{(x-a)^{n+1}(x-b)^{n-1}}}$$
0
votes
3answers
19 views

Evaluate integral with Partial fraction expansion

i'm having problem with my equation-system in this task. $\int \frac {1-7x}{(x^2 + 1)(x-3)} \text{d}x$ Done this: $\int \frac {Ax + B}{(x^2 +1)} + \frac {C}{(x-3)} \text{d}x $ And got this ...
3
votes
2answers
52 views

Integrate $\int \frac{dx}{(x^2+c)^\frac{3}{2}}$

Using Mathematica, I found a simple result $$ \int \frac{dx}{(x^2+c)^\frac{3}{2}} = \frac{x}{c\sqrt{x^2+c}} + const$$ where $c$ is a constant. But I am unable to get this result by hand - I don't ...
1
vote
1answer
31 views

Numerical integration of function - result is another function

I'm new in integration and numerical integration. As i know to calculate definite integral you can use some methods, like rects, trapezes, Simpson's... etc. But is there a tool to make numerical ...
5
votes
0answers
48 views

Integrating $I=\int\frac{x}{\sqrt[4]{x^3(a-x)}}dx, a>0$

This is the integral I've come across while solving the workbook: $$I=\int\frac{x}{\sqrt[4]{x^3(a-x)}}dx, a>0$$ My solution isn't the same as one in the workbook, so, please, tell me where I made a ...
3
votes
3answers
55 views

Indefinite integral of $ \int \frac{x^3}{\sqrt{x^2+1}} \,dx$

Can you please provide any sort of hint or suggestion on how to find the indefinite integral of $$ \int \frac{x^3}{\sqrt{x^2+1}} \,dx $$ ...I tried substituting everything but it didn't work. I also ...
0
votes
3answers
28 views

What is the alternating sum of coefficients and what does it have to do with the zeroes of the function?

So, my teacher told us today, while we were solving this integral: $$\int\frac{dx}{x(2x^3+x^2+1)}$$ that the alternating sum of coefficients of $2x^3+x^2+1$ is 0 (2-1+0-1=0) and hence, one zero of the ...
2
votes
0answers
77 views

$ \int \frac{\sin^{a}x}{\sin^{a}x+\cos^{a}x}dx$ and $ \int \frac{\cos^{a}x}{\sin^{a}x+\cos^{a}x}dx$

I tried solving this integral: $$\int^{\pi/2}_{-\pi/2} \frac{1}{2007^x + 1}\frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$ I took a while before aptly applying the following identity I had noted ...
0
votes
2answers
75 views

Finding antiderivitive of $\frac{x}{\sqrt[3]{1-3x}}$

Evalute: $\int \frac{x}{\sqrt[3]{1-3x}}dx$ my try: $u=1-3x$, $x=\frac{1-u}{3}, dx = - \frac{1}{3}du$ $\int \frac{x}{\sqrt[3]{1-3x}}dx = \int \frac{1-u}{3} \frac{1}{\sqrt[3]{u}}(-\frac{1}{3})du = ...