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

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4
votes
2answers
68 views

Evaluate $\int \ln(1 + e^x)\ \mathrm dx$

Evaluate the following indefinite integral. $$\int\ln(1 + e^x) \mathrm dx$$ My attempt :: Using integration by-parts, \begin{align} \int\ln(1 + e^x)\cdot 1\ \mathrm dx &= x\ln(1 + e^x) - \int ...
5
votes
2answers
58 views

Quadratic Expressions: Advanced techniques of Integration

$$\int \frac{x}{\sqrt{5+12x-9x^2}}\,dx$$ After two steps I arrive at $\displaystyle{ \int \frac{x}{\sqrt{9-(3x-2)^2}}}\,dx$ Using trigonometric substitution, we have a triangle with a cosine of ...
3
votes
3answers
61 views

Find $ \int \frac {1-x^2}{1+3x^2+x^4} \, \mathrm{d}x $

Today, the CalcBee sample problems got released. The following problem was my creation and I wanted to see how many solutions people can come up with. The result is very beautiful and I thought it ...
5
votes
4answers
118 views

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

Evaluating $$\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$$ using $ux=\sqrt{x^2-1}$ I try to $u^2x^2=x^2-1$ $x^2=\frac{-1}{u^2-1}$ However I cant get rid of $x$ because derivative has $x\;dx$. How can I ...
2
votes
1answer
34 views

Why $ \int_0^{\infty} du \, \frac{e^{-3 u} - e^{-4 u}}{u} = \int_0^{\infty} du \, \int_3^4 dt \, e^{-u t} \\ $?

from this answer I could not see what is happening here: $$ \int_0^{\infty} du \, \frac{e^{-3 u} - e^{-4 u}}{u} = \int_0^{\infty} du \, \int_3^4 dt \, e^{-u t} \\ $$ What technique of integration ...
1
vote
3answers
67 views

Seemingly Simple Integration: $x/(x-1)$

I am currently working on some advanced engineering math but this seemingly simple integral has me stuck. Someone please show me how to derive it. It is part of a far bigger more complex problem in ...
1
vote
3answers
48 views

Evaluate $\int {x+3\over x^2+6x+10}dx$ [on hold]

$$\int {x+3\over x^2+6x+10}dx$$ Could anyone help me with this substitution problem?
1
vote
1answer
53 views

How to evaluate $\int \cot^2(x) \;\mathrm dx$?

How do you find the antiderivative of $\cot^2x$? My steps to find it First $$ \csc^2 x = \cot^2 x+ 1 $$ because of Pythagorean Identities, so $$ \cot^2 x= \csc^2 x-1$$ so $$ \int \cot^2 x\, ...
0
votes
2answers
44 views

What's $\int \frac{1}{\sqrt{25-x^2}}$ [duplicate]

What is $$\int \frac{1}{\sqrt{25-x^2}}$$ WolframAlpha says $\sin^{-1}(\frac{x}{5})$ while I got $\frac{1}{5}\sin^{-1}(\frac{x}{5})$. What is correct? Thanks in advance.
1
vote
1answer
21 views

Volume of a solid formed as vertical limit goes to infinity

Here's the question: The way I have it set up currently is as follows: $V = \pi \lim_{a \to \infty} \int_1^a (a-1)^2 - (\frac{1}{\sqrt{x^5}} - 1)^2$ But how do I go from here? And is the working ...
5
votes
2answers
87 views

Antiderivative of $\frac{1}{\ln(x)}$?

I was looking on wikipedia, and found that the following expression cannot be expressed in terms of elementary functions: $$\int\frac{1}{\ln(x)}\text{d}x$$ Although the function looks simple, why is ...
14
votes
4answers
216 views

Evaluate $\int_1^\infty \frac {dx}{x^3+1}$

I would like some help with the following integral. I would like to find a contour line to evaluate $$\int_1^\infty \frac {dx}{x^3+1}$$ So one can see that on any circumference it goes to $0$, but ...
4
votes
4answers
57 views

Evaluate $\int\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm dx$

I have tried to evaluate $$∫\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm d x$$ using the following identity: $$\frac{d(\sin^{-1}{u})}{du} = \frac{du}{1+u^2}$$ So I then reformed the integral to this: ...
1
vote
3answers
72 views

Help evaluating $\int e^x \sqrt{1+e^{2x}}dx$ [duplicate]

$\int e^x \sqrt{1+e^{2x}}dx$ It's probably been answered somewhere, but I havent found it so far so I decided to post it as a question (if it has been answered point me in the right direction and I ...
2
votes
2answers
65 views

Evaluation of $\int\frac{1}{1+(x+1)^{{1}/{n}}}dx$ for $n\in \mathbb{N},$

Evaluate $$\int\frac{1}{1+(x+1)^{{1}/{n}}}\,\mathrm dx$$ for $n\in \mathbb{N}$ $\bf{My\; Try::}$ Let $$(x+1)=t^n\;,$$ Then $$dx = nt^{n-1}dt$$ So $$\displaystyle I = ...
18
votes
2answers
994 views

Is indefinite integration non-linear?

Let us consider this small problem: $$ \int0\;dx = 0\cdot\int1\;dx = 0\cdot(x+c) = 0 \tag1 $$ $$ \frac{dc}{dx} = 0 \qquad\iff\qquad \int 0\;dx = c, \qquad\forall c\in\mathbb{R} \tag2 $$ These are two ...
4
votes
0answers
67 views

What is the indefinite integral of zero? [duplicate]

From the definition of indefinite integral I might say: Since the derivative of a constant is zero, thus the indefinite integral of zero is a constant. Therefore: $$ \frac{dc}{dx} = 0 \quad\iff\quad ...
0
votes
1answer
52 views

Who introduced the term indefinite integral and the notation $\int f(x)dx$?

I find the notation $\int f(x)dx$ for the indefinite integral of $f(x)$ on some interval $I$ is both suggestive and confusing. On the one hand, this notation is very suggestive when we calculate ...
4
votes
2answers
70 views

How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$?

I am looking to demonstrate the following result. Any ideas are much appreciated. $$ \begin{align}\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx = &\;x \ln \left( b + \sqrt{b^2 +c^2 ...
3
votes
2answers
90 views

The integral of $e^{-x^2}$ [duplicate]

How can I integrate this by parts? It seems to become recursive. I'm familiar with the classical solution, and cannot use that here due to the constraints of this class. Here's the integral (to ...
-1
votes
1answer
34 views
3
votes
5answers
95 views

Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $

Regarding the integral $$ \int \frac{dx}{x^2\sqrt{x^2 + 1}} $$ I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?
-1
votes
1answer
86 views

Evaluate $\int\left(1+x^2\right)e^{\arctan x}\,dx$ [closed]

How does one evaluate the following integral $$\int\left(1+x^2\right)e^{\arctan x}\,dx$$ Thanks.
0
votes
2answers
74 views

Evaluating $\int \frac {\ln(3x+7)}{x^2}\mathrm dx$

How to evaluate the following integral? $$\int \frac {\ln(3x+7)}{x^2}\,\mathrm dx$$ I tried substituting both $u = 3x+7$ and $u = \ln(3x+7)$, but the resulting integral seems to be much more ...
4
votes
2answers
74 views

Evaluate $\int\frac{\cot x}{\cos^2 x-\cos x+1}\,\,dx$

$$\int\frac{\cot x}{\cos^2 x-\cos x+1}\,\,dx$$ Please guide me by which term it should be substituted to get the result of this integration. I have tried it by using $\cos x =t$, but it went so long ...
4
votes
1answer
171 views

Evaluating $\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$

The title says it all, really - I am looking for $$\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$$ where $0<r<1$ and $x$ is in a domain where the integrand is real. It came up ...
1
vote
3answers
59 views

Evaluating $\int{\frac{du}{3e^{u}+1}}$

why is $$\int{\frac{du}{3e^{u}+1}}=\ln\frac{e^u}{3e^u+1}+c$$ ? I think some substitution should help solving this integral, but everything I tried did not work.
0
votes
0answers
13 views

Do these “algebraically well behaved” Function Spaces, exist?

Do there exist any Sets of Functions which are some combination of: Algebraically Closed: In the sense of Algebraic Functions. Differentially Closed: In the sense of Differentially Closed Fields. ...
0
votes
1answer
20 views

Tedious differential equation, solving for when $y(t) = 0$

If I have the differential equation: $$\begin{cases} \pi(2\,y\,R-y^2)\frac{dy}{dt}=-\pi\,r^2\sqrt{2\,g\,y} \\ y(0) = R \\ y'(0) < 0 \end{cases}$$ Where $r, R,$ and $g$ are all constants. If I ...
1
vote
1answer
42 views

Integral of power mean function

I wonder, is there a closed-form solution to integral of type $$\int \left(\frac{a^x+b^x}{2}\right)^{1/x}dx,$$ for $a\ne b$ (both positive real numbers). If not, what about a special case when $a=1$, ...
1
vote
2answers
23 views

Evaluating $\frac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$

$$\dfrac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$$ The answer I come up with is: $5\sin(3x)(3)-7e^{4(3x)}(3)$, however this was not on the answer choice. What is the ...
-1
votes
1answer
56 views

Evaluating $ \int \frac{1}{\sin x} dx $

Verify the identity $$\sin x = \frac {2 \tan\frac{x}{2}}{1 + \tan^2\frac{x}{2}}$$ Use this identity and the substitution $t = \tan\frac{x}{2}$ to evaluate the integral of $$ \int \frac{1}{\sin x} dx ...
2
votes
1answer
34 views

Evaluate $ \int \frac{\tan(x)}{2+\sin(x)}dx $

How do you evalute this integral? $$ \int \frac{\tan(x)}{2+\sin(x)}dx $$
2
votes
2answers
44 views

Integral of $\dfrac{\cos(x)}{5+3\cos(x)}$

I was doing $$\int\!\mathrm{d}x \dfrac{\cos(x)}{5+3\cos(x)}$$ and using the substitution $\cos(\theta) = \dfrac{1-t^2}{1+t^2},\quad t = \tan\left(\dfrac{\theta}{2}\right)$ ...
4
votes
4answers
305 views

Evaluate $\int\sqrt[5]{\frac{x+5}{x-5}}\,\mathrm dx$

How to compute this integral? I stuck at a point where I get $\displaystyle\int\frac{1}{t^5-1}+ \cdots $ $$\int\sqrt[5]{\frac{x+5}{x-5}}\,\mathrm dx$$ using $\displaystyle t=\sqrt[5]{\frac{x+5}{x-5}}$ ...
7
votes
0answers
85 views

How to Evaluate $\int\frac1{x \ln x+ 7 \ln x} \,\mathrm dx$

I have tried many methods but do not know how to integrate this: $$ \int \frac{1}{x\ln x + 7\ln x} dx $$ with respect to x.
2
votes
3answers
60 views

Evaluating $\int\sec x \,\mathrm dx$ [duplicate]

$$\int\sec x \,\mathrm dx = \ln\left|\sec{x} + \tan{x}\right|+ C = \ln{\left|\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\right|} + C$$ My question is how? How are these derived?
4
votes
1answer
40 views

How to integrate $\int \frac{1}{\sqrt{1+29x^2+100x^4}}dx$ and $\int \frac{1}{\sqrt{1-2x^2-8x^4}}dx$ using elliptic functions?

How to integrate $$\int \frac{1}{\sqrt{1+29x^2+100x^4}}dx$$ and $$\int\frac{1}{\sqrt{1-2x^2-8x^4}}dx$$ using elliptic functions? I have tried to use them, but I got incorrect formula ...
4
votes
4answers
117 views

Evaluating $\int \frac{\sec^2 x}{(\sec x + \tan x )^{{9}/{2}}}\,\mathrm dx$

How do I evaluate $$\int \frac{\sec^2 x}{(\sec x + \tan x )^{{9}/{2}}}\,\mathrm dx$$ I've started doing this problem by taking $u=\tan x$.
1
vote
1answer
37 views

Partial Fractions Variables

$$\int\frac{x^3+6x^2+3x+16}{x^3+4x}\,dx$$ Eventually one solves that a variable (I used $C$) ${}= -x$. By variables I mean the decomposition yields $A + Bx+C$. Therefore $C = -1$. I think that $C$ ...
2
votes
3answers
44 views

Integrating trigonometric functions.

I need a little help with solving the following two integrals. I am not able to approach these two problems correctly. Little hints will suffice. Thanks for your help! $$\int\frac{\tan^2x\sec^2x}{1+ ...
0
votes
2answers
37 views

How many days until the population doubles

I have a function called $P(t)$ that is the number of the population at time $t$. $t$ being in days. We know the growth rate is $P'(t) = 2t + 6$ We also know that $P(0) = 100$. How many days till ...
2
votes
1answer
40 views

Indefinite integral of absolute value

When I looked up about absolute value on Wikipedia, I found that the antiderivative of $|x|$ is $\frac12 x|x|+C$. I am able to find the derivative of $|x|$ by treating the function as $\sqrt{x^2}$, ...
2
votes
1answer
54 views

Integral of rational function $\int\frac{x^2-18x-1}{(x-1)^2(x^2+1)} \text{d}x$

Evaluate $$\int\dfrac{x^2-18x-1}{(x-1)^2(x^2+1)} \text{d}x$$ (Remember to use $\ln |u|$ where appropriate. Use $\text{C}$ for the constant of integration.) I got the answer $$\dfrac{-1}{2 ...
3
votes
2answers
74 views

Evaluate $\int \ \frac{2}{{}x\sqrt{9x^2 - 25}} dx$

I'm trying to evaluate $$\int \ \frac{2}{{}x\sqrt{9x^2 - 25}} dx$$ So I know that if I had just $$\int \ \frac{2}{{}x\sqrt{9x^2 - 25}} dx$$ then I would be able to use a natural log rule ...
4
votes
3answers
124 views

How difficult exactly is $\int\tan(x^2)\ dx$?

How difficult exactly is $\int\tan(x^2)\ dx$ ? Is it possible to express this integral in terms of elementary functions? If not, is there anything one could say about it, that would be in some way ...
1
vote
1answer
48 views

Incorrect indefinite integral on MATLAB?

any ideas why matlab is giving me an incorrect answer here? 1 set of commands syms x L N2 N2 = 6*x/L^2 - 2/L^2 N2 = (6*x)/L^2 - 2/L^2 expand(int(N2*N2)) ans = ...
0
votes
1answer
24 views

$f'(x)=(3/x)-4; (1,0)$ derivative at a specific point

So I have been trying to figure out how to do this type of problem for quite a while now. Question States: The slope $f'(x)$ at each point $(x,y)$ on a curve $y=f(x)$ is given along with a ...
0
votes
1answer
42 views

Evaluate $\displaystyle\int-x^{1-n}e^{xt}\ dx$

I have to evaluate $$\large\displaystyle\int-x^{1-n}e^{xt}\ dx$$ with respect to x but I am not sure how. I have tried integration by parts but this gets very complicated, is there an easier way? ...
2
votes
2answers
48 views

Evaluate $\int\frac{dx}{x(2-\ln x^2)}$

How to evaluate $\displaystyle\int\frac{dx}{x(2-\ln x^2)}$? I tried to use an operator, but before that I distributed so $\displaystyle\int\frac{dx}{(2x-x\ln x^2)}$ my operator was $-x\ln x^2$ am I ...