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

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1
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3answers
46 views

Trigonometric integral evaluation: $\int 4 \sin^4 x \cos^3 x \,dx$

Evaluate the following integral $$\int 4 \sin^4 x \cos^3 x \,dx$$ I can do simple integration problems, but problems like this seem to stump me, I created this problem so I could solve and compare it ...
1
vote
3answers
46 views

I need help solving this indefinite integrals problem?

I am doing indefinite integrals homework, and this problem popped up. I hate to post on here without any personal insight on the problem, but I really have no idea on how to approach this.I do not ...
2
votes
1answer
64 views

Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $

Evaluate the indefinite integral $$\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx$$ I tried multiplying and dividing by $\sec^2 {x} $ and then setting $\tan{x}=y$ but no good. Then I set $\cos ...
1
vote
1answer
73 views

Is this formula for the harmonic numbers true?

Is this formula for the harmonic numbers true? $$H_n = \lim_{s\to 0} \, \int \frac{(s+1)^{(-n-1)}+s-1}{s} \, ds$$ Mathematica: ...
3
votes
4answers
127 views

How would I go about evaluating $\int \frac{x}{(9-8x^2)^3}dx$?

So I have homework on webAssign (a site used by my college), and I am not understanding the logic as to why I am taking the steps into solving the integral it is telling me to take. So I'll list the ...
0
votes
2answers
42 views

What is happening to the '2' in this integral?

It is the indefinite integral: $\int \frac{1}{2x-6}$ I am trying to understand it and looking the last step goes from $\frac12 \log(2(x-3))$ to $\frac12 \log(x-3)$ Can someone explain to me why the ...
0
votes
3answers
86 views

Evaluating $\int x^2 \sqrt{x^2-1} dx$

How do I evaluate the following indefinite integral? $$\int x^2 \sqrt{x^2-1} dx$$ Through integration of parts, I have obtained $$ \frac{x}{3}(x^2-1)^{3/2} - \frac{1}{3} \int (x^2-1)^{3/2} dx $$ ...
0
votes
0answers
26 views

Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
-1
votes
0answers
25 views

changing the variable in integration [duplicate]

What is the way to change the variable with respect to we are doing integration For example in this integral [(cos2x)^1/2]dx/sinx How to change dx to d(cosx)
2
votes
3answers
180 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
1
vote
4answers
486 views

How useful/useless is the indefinite integral

After having met yet another person confused by indefinite integrals today, I've finally decided to ask the community. Do you think it makes sense to teach indefinite integrals? My opinion is that ...
0
votes
0answers
56 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
0
votes
0answers
57 views

Is the following integration of possible?

How to solve the following problem? $\int x^4/(1-x^4)^{3⁄2}dx$ I have tried the substitution, x=sinz, but failed.
2
votes
4answers
96 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
4
votes
2answers
119 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...
1
vote
2answers
72 views

How do I find this integral $\int\frac{dx}{2x^4+2x^2-1}$

How I evaluate the above integral? $$\displaystyle\int\dfrac{dx}{2x^4+2x^2-1}$$ I have unsuccessfully tried it more than once. Is there a small substitution that I am missing? And is there any ...
4
votes
1answer
117 views

How can I evaluate this indefinite integral? $\int\frac{dx}{1+x^8}$

How do I find $\displaystyle\int\dfrac{dx}{1+x^8}$? My friend asked me to find $\displaystyle\int\dfrac{dx}{1+x^{2n}}$ for a positive integer $n$. But looking up I am getting pretty noisy answer for ...
2
votes
2answers
32 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
2
votes
2answers
147 views

integrate $ \int \frac {x dx}{\sqrt {1+x^{10}} } $

This is a tough one. Thanks. $$\int \frac {x dx}{\sqrt {1+x^{10}} } $$ This is not a homework problem. I spend 10 hours over the course of 3 days on this. I tried: 1) substituting u for x^5 to get ...
2
votes
4answers
127 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
1
vote
1answer
47 views

Equation with integral

I have the following equation: $$\int (x-b)^n(x-c)^mdx = \frac{f(x)}{a}.$$ I want to compute value of $a$, but I don't know how can I escape this integral. $b$, $c$, $n$, $m$ are constants.
2
votes
2answers
56 views

Integration by Tables problem

$$\int \frac {dx} {x(x^8-256)}$$ I am supposed to use the formula $$\int \frac {dx} {x(ax+b)} = \frac1b\ln\left|\frac x {ax+b}\right|+C $$ to find the integral. I don't know how to start. Help is ...
1
vote
2answers
45 views

Doubt in integral substitution

I am not able to figure out what substitution to use in the following integral $$ \int \frac{(x-1)e^x}{(x+1)^3}dx $$ Any help would be appreciated.
3
votes
1answer
48 views

Does $\int { y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$ have a closed form

I am trying to solve the following indefinite integral $$F_Y(y) = \int {y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$$ Where $J_0$ is the Bessel function of the first kind. I tried ...
4
votes
5answers
83 views

Integration problem $\displaystyle \int \frac{dx}{x(x^3+8)}$

$$\int \frac{dx}{x(x^3+8)}$$ I think I'm supposed to use partial fractions, but I am unsure of how to start the problem. Any help would be appreciated.
1
vote
3answers
92 views

Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}dx$

How to do this indefinite integral (anti-derivative)? $$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}dx$$ I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
0
votes
0answers
36 views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
-1
votes
1answer
37 views

Proving the indefinite integral $ \int \frac{1}{u^2(a+bu)}du $ [closed]

How can I prove that the indefinite integral $$ \int \frac{1}{u^2(a+bu)}du $$ is equal to $$ -\frac{1}{a}\left(\frac{1}{u}+\frac{b}{a}\ln\left|\frac{u}{a+bu}\right|\right)+C\ ? $$
1
vote
3answers
109 views

Evaluate $\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$

$$\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$$ My approaches so far has been using substitution with $\tan x = t$ and $\tan \frac x2 = t$ but the calculations has been harder than I ...
1
vote
3answers
63 views

Steps to solve $\int \sqrt{\frac{11}{x}}\,\mathrm{d}x$?

What are the steps required to solve the following? $\int \sqrt{\frac{11}{x}}\,\mathrm{d}x$ I'm not looking for anyone to do my homework. I usually have no problem figuring these things out -- ...
3
votes
1answer
77 views

Integrate : $\int(\sin x+\cos x)^ndx$

Problem : $$\int(\sin x+\cos x)^n\ dx$$ I am not getting any clue how to integrate this. Please help . I will be grateful to you. Thanks.
2
votes
1answer
47 views

Solving indefinite integrals gives multiple answers. Are all those answers correct?

While solving problems on indefinite integrals many a times I get answers which are different from those given in my text book's answer keys page. I then verify my solution steps to ensure that even ...
5
votes
1answer
93 views

Determining the best possible constant $k$, for an Integral Inequality

If $f : [0,\infty) \to [0,\infty)$ is an integrable function, then what is the best possible constant $k$, for which the following ineqality holds: $$\int_0^{\infty}f(x)dx \leq ...
7
votes
3answers
113 views

Evaluate $\int {x \choose n} \ dx$ (Problem 798 Crux Mathematicorum)

Evaluate $$I_{n}= \int {x \choose n} \ dx$$ where $n$ is a non-negative integer.Any idea of what closed form $I_{n}$ will have.
-1
votes
1answer
59 views

sin x integral qestions [duplicate]

How could the following integral be solved in a good manner? $$\int \frac{\sin(x)}{x}\;\mathrm{d}x$$ Regards:
1
vote
1answer
32 views

missing $j*\omega$ in integral

let us consider following integral according to property of delta function,we can write this intgeral as $\int^{t=\infty}_{t=t_0} e^{-j*\omega*t}$ or we can write as ...
2
votes
4answers
72 views

indefinite integral computation $dx/(e^{-x}-x)$

Hi i'm trying to carry out the following indefinite integral: $$\int \frac{1}{e^{-q} - q} \, dq$$ mathematica is not helping me, and i think it is not solvable by substitution method. any idea on ...
0
votes
0answers
58 views

Mathematica Integrate gives back the integrand

i'm trying to Integrate the following function: (q (1 + q) - E^-q Sinh[q])/(-q + Cosh[q] Sinh[q]) - ( 2 q Tanh[q])/(-q + Cosh[q] Sinh[q]) I already solved ...
6
votes
3answers
451 views

Are indefinite integrals unique up to the constant of integration?

We often write e.g. $$\int x^2 dx=\tfrac{1}{3}x^3+c$$ for any $c \in \mathbb{R}$, where $c$ is the constant of integration. We can show (via limits) that, if $g(x)=\frac{1}{3}x^3+c$, then ...
0
votes
0answers
18 views

Sufficient condition for a indefinite integral to be an elementary function

I would like to find a sufficient condition on two polynomials $P(s)$ and $Q(s)$, such that the function $s \mapsto Q(s)e^{P(s)} $ has a primitive integral of the form $s \mapsto R(s)e^{P(s)} $ (with ...
0
votes
1answer
60 views

Given $f_X$. Integrate $\int_0^\infty \log_2 (x+1) f_X \, dx$.

Say $Y=Log_2[1+x]=g(X)$ and $f_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ ...
4
votes
1answer
63 views

Determine the indefinite integral $\int \frac{\sin x}{\sin 5x}dx$ [closed]

Find the value of the given integral $$\int \frac{\sin x}{\sin 5x}dx$$
2
votes
3answers
116 views

How to evaluate the following integral? $\int \frac{x^6}{x^4-1} \, \mathrm{d}x.$

Evaluate the integral: $$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$ After a lot of help I have reached this point: $x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$ But now I ...
2
votes
1answer
59 views

Strange error concerning integration by parts

First, this is not homework; I just decided to try a classic integral in a non-standard way and came out with a strange result. The integral $I:=\int\frac{dx}{x\ln x}$ is well-known to equal $\ln\ln ...
8
votes
4answers
145 views

How to calculate $\int\frac{1}{x + 1 + \sqrt{x^2 + 4x + 5}}\ dx$?

How to calculate $$\int\frac{1}{x + 1 + \sqrt{x^2 + 4x + 5}}dx?$$ I really don't know how to attack this integral. I tried $u=x^2 + 4x + 5$ but failed miserably. Help please.
0
votes
2answers
65 views

Improper integral with removable discontinuity

Integrate , for $ \alpha > 2 $ $ \int_0^{\infty}\!\frac{x-1}{x^\alpha-1}\, dx. $ I would be intertest for any replies or any comments
4
votes
4answers
93 views

Evaluation of $ \int \tan x\cdot \sqrt{1+\sin x}dx$

Calculation of $\displaystyle \int \tan x\cdot \sqrt{1+\sin x}dx$ $\bf{My\; Try::}$ Let $\displaystyle (1+\sin x)= t^2\;,$ Then $\displaystyle \cos xdx = 2tdt\Rightarrow dx = ...
2
votes
2answers
127 views

Compute the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$ or $I=\int_{d}^1 y^{-a}(1−y)^{b-1} dy$

I need to calculate the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$, where $a$, $b$ are REAL NUMBERS and $b>0$. (my goal is to determine the definite integral $I=\int_{d}^1 y^{-a}(1−y)^{b-1} ...
2
votes
2answers
27 views

how to calculate integrate about Heaviside

everyone,here I have a question about how to calculate $$\int e^t H(t) dt$$ where $H(t)$ is Heaviside step function thank you for your answering!!
1
vote
2answers
70 views

Indefinite integral of $x^x$

I've seen many many questions on the internet with answer that it cannot be done with elementary functions. Now I did this integration myself and got a pretty nice result. Since I've seen so many ...