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

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4
votes
2answers
78 views

Finding $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$

I wanted to find $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$, so I substituted $t=\sin\theta$ and got $\int\frac{\cos^2\theta}{1+\sin^2\theta}d\theta$; but I'm not sure what the best way to proceed from here ...
2
votes
0answers
12 views

double integration with the same variable

I have the integral that I want to resolve. To calculate the flux of the electric machine, I have the following formula: $v_s= R_s \cdot i_s + \frac{\Phi _s}{dt}$ where $v_s, i_s, \Phi _s$ are ...
5
votes
4answers
101 views

Finding $\int\frac{1}{x^{11}+4x^6}dx$

I wanted to find out if there is an easy way to evaluate $\displaystyle\int\frac{1}{x^{11}+4x^6}dx$. I substituted $u=x^5$ and then used partial fractions, but maybe there is a simpler way to find ...
3
votes
0answers
45 views

What is $\int \frac{e^{a x}}{1+x^2} dx $?

In my answer to another question (here: Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$), I came up with this integral: $\int \frac{e^{a x}}{1+x^2} dx $. I ...
6
votes
6answers
89 views

Two apparently different antiderivatives of $\frac{1}{2 x}$

What is right way to calculate this integral and why? $$ \int\frac{1}{2x}\text dx $$ I thought, that this substitution is right: $$ t = 2x $$ $$ \text dt = 2\text dx $$ $$ \frac{\text dt}{2} = ...
2
votes
2answers
65 views

Find the value of undefinite integral

Find $$\int \frac{dx}{(x+1)^{1/2}+(x+1)^{1/3}}$$ I have tried with let $u=(x+1)^{1/2}+(x+1)^{1/3}$ but I have nothing to solve that undefinite integral. please give me a clue for solve it.
9
votes
3answers
82 views

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

I'm given this $\int\frac{1}{\sin x+\cos x}dx$. My attempt, $\sin x+\cos x=R\cos (x-\alpha)$ $R\cos \alpha=1$ and $R\sin \alpha=1$ $R=\sqrt{1^2+1^2}=\sqrt{2}$, $\tan\alpha=1$ ...
1
vote
0answers
22 views

Integral of a binomial with non-integer exponent

I need to find a closed form or at least get a fair approximation (with a series expansion) of the following integral: $$\int \left( Ax+\frac{B}{x} \right)^{1/n}\,dx$$ Where $n$ is any integer and ...
-1
votes
0answers
38 views

Integral: $\int \frac{1}{1+x^4}dx$ [duplicate]

I asked my teacher how to do it, he answered I had to use series but I won't learn this method in highschool. But I want to know how to solve it. I read online about series but I don't see how ...
2
votes
4answers
339 views

Strange solution for the integral

Reading some examples in my theory book, I met strange solution and I can't figure out how did they got it. Here it is $$\int \frac{v \, dv}{\sqrt{v^4 +g^2r^2}}=\frac{1}{2}\ln{ ...
2
votes
1answer
64 views

Arc length of natural log function

I am currently trying to find the arc length of $f(x)=ln(x)$, which involves the integral $$\int \sqrt{1+\frac1{x^2}}dx$$ I managed to solve the integral correctly but I want to know if there is a ...
2
votes
0answers
22 views

Indefinite Hypergeometric Integral Transformations

I'm attempting to solve the indefinite integral $$S\left(v\right) = 2a\sqrt{\alpha ...
3
votes
2answers
44 views

What is indefinite $ \int\frac{\sin\left(x\right)\cos\left(x\right)}{\sqrt{3 - x^{4}}}{\rm d}x . $

$$ \int\frac{\sin\left(x\right)\cos\left(x\right)}{\sqrt{3 - x^{4}}}{\rm d}x . $$ Hello. Today, in exam, we had to evaluate this integral. Noone was able to do it. Appreciate any help.
2
votes
3answers
74 views

How evaluate $\int \frac{\cos^2(x)}{1 + \text{e}^x}dx$ to find an improper integral

Can someone help me evaluate this: $$\int \frac{\cos^2(x)}{1 + \text{e}^x}dx\;?$$ I need it for determining whether the improper integral $\int_0^\infty {\frac{{\cos^2{{(x)}}}}{{1 + ...
0
votes
0answers
13 views

Mean value theorem for an indefinite integral

I want to solve the following integral (which has to depend on $u$) $$I(u)=\int_{u_1}^u \frac{f(x)}{x(1-x)}dx,~~~~~~u_1=const$$ I would like to know if I can apply the mean value theorem like this: ...
0
votes
3answers
68 views

Evaluate the integral $\int \frac{dx}{x^3 + 2x^2 + 2x}$ of a rational function

Evaluate $$\int \frac{dx}{x^3 + 2x^2 + 2x}.$$ I have no idea how to approach this. I know how to solve rational functions with numerator as highest degree polynomial using division and remainder. ...
-6
votes
1answer
38 views

What is a triple integral with the answer of 39? [closed]

What is a triple integral with the answer of 39
0
votes
2answers
27 views

Why does this relationship hold (integrals)

If I define $$P(t,T) = \exp\{ \int _t^Tf(t,s)ds\}$$ then why is it true that $$\ln P(0,T) - \ln P(0,t) = - \int_t^T f(0,s)ds \tag{1}$$ We would have $$P(0,T) = \exp \{ - \int_0^T f(0,s)ds\}$$ ...
2
votes
0answers
57 views

$\int \frac{1}{x^2(1+x)(1-x)^{5/3}}dx$

I am solving this differential equation: $$(1-x^2)y'' -2xy' + 2y = 0$$ Noticing that $x$ is a particular solution, we may use the method of reduction of order. Letting $y = xz$, this becomes: ...
12
votes
1answer
126 views

An integral with three radicals in the denominator, $\int \frac{\mathrm{d}x}{\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}}$

$$\int \frac{\mathrm{d}x}{\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}}$$ I tried substituting $x=z^2$ ... also $x=\tan^2 \theta$ ... but couldn't solve it either ways... if someone can help then it would be good. ...
4
votes
2answers
95 views

Calculate $\int \limits {x^n \over 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+…+\frac{x^n}{n!}} dx$ where $n$ is a positive integer.

Calculate $$\int \limits {x^n \over 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}} dx$$ where $n$ is a positive integer. Would you give me a hint?
2
votes
0answers
35 views

Integration involving exponential function

I am trying to find the following integral $\int_t^s e^{-\frac{(\sqrt{x^2-a^2}-u)^2}{\sigma^2} } e^{-\frac{b^2}{x^2}} dx$ where $t<u<s$, I can find a solution for $u=0$, but I need the ...
2
votes
2answers
73 views

Solve the integral $\int \frac{dx}{\:\sqrt[4]{\left(x+2\right)^5\cdot \left(x-1\right)^3}}$ [closed]

Here is a indefinite integral must be solved. Help, who knows. Although it would be like casual. $$\int \frac{dx}{\:\sqrt[4]{\left(x+2\right)^5\cdot \left(x-1\right)^3}}$$
1
vote
0answers
22 views

division of two integrations

I am new to calculus. Today when I read the exponential family, The exponential family are defined as below: $$ p(x|\alpha) = h(x)exp\{ \alpha T(x) - A(\alpha)\}$$ $ T(x) $ is referred to as ...
1
vote
1answer
68 views

Evaluating $\int \frac{x^2}{1+e^{-x}}dx$

Consider $$\int \frac{x^2}{1+e^{-x}}dx$$ I've tried every method and trick that I'm familiar with, except by parts, but I can't seem to be able to acquire an elementary integral. Does there exist ...
-1
votes
1answer
27 views

Integral $\int\frac1{2x(1+\sqrt x)}\,\mathrm dx$ and limit $\lim\limits_{x\to\infty} (3^x+3^{2x})^{1/x}$

What is $\int\frac1{2x(1+\sqrt x)}\,\mathrm dx$? I have tried this using substitution but nothing worked. I know substitution can solve this but the correct substitution is just not ...
2
votes
4answers
301 views

There is some strategy to solve an integral of this kind?

How to solve the integral $$\int\frac{\ln x}{\sqrt{1-x}}dx$$ and $$\int\sqrt{\frac{x}{x-1}}dx$$ I have no idea of how to deal with these integrals. It's the first integral I attempted. ...
0
votes
0answers
33 views

How to convert infinite intergral to sum

How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated: I could not find much info on ...
-2
votes
1answer
38 views

Find the indefinite integral and check the result by differentiation

Find the indefinite integral and check the result by differentiation. I have worked all the problems just I am stuck and would like to check my answers. (1) $\int(x-x^2)dx$ (2) ...
3
votes
1answer
38 views

Solving $\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right)$

I got as far as: $$\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right) $$ $$=\frac{2}{\sqrt{\pi}} \int dx \int^{c(x-b)}_0 dy {\sqrt{x^2+a}} e^{-A x^2 - y^2}$$ $$=\frac{-2 c}{\sqrt{\pi}} ...
1
vote
0answers
22 views

Need help with integrating the function $\frac{-cb}{b(ax+Pbe^{bx})+a}$ w.r.t. $x$

Can someone give me some hints on how to solve the following: $$exp\left(-\int{\frac{cb}{b(ax+Pbe^{bx})+a}\,dx}\right),$$ where $exp(t)=e^{\,t}\,\forall t\in \mathbb{R},e \approx2.71$, and $a,b,c,P$ ...
1
vote
2answers
74 views

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

$$\int \frac{1}{(x+\sqrt{x+x^2})^2} dx$$ I don't know how to approach this integral. I tried a few substitutions, but none of them got me to a desirable point.
1
vote
1answer
82 views

Integral of $ \int \frac{x}{\sqrt{4x^2 + 8x + 5}} dx$

How to solve: $$\int \frac{x}{\sqrt{4x^2 + 8x + 5}} dx$$ This question is from a list and it's in the category of problems that involving $\sqrt{x^2\pm a^2}$ and $\sqrt{a^2\pm x^2}$ (triangle ...
2
votes
1answer
34 views

How to evaluate the integral $\int_{0}^{2}{g(x) dx}$, where $g(a)$ is a solution of the equation $x^{5}+x=a$

We consider an integral $\int_{0}^{2}{g(x)dx}$, where $g(a)$ is a solution of $x^{5}+x=a$. Actually, it means that $g^{5}(a)+g(a)-a=0$. Moreover, it somehow possible to reestablish $g(x)$ on $[0, 2]$ ...
0
votes
3answers
28 views

Find the indefinite integral

$$\int \frac{x^3 - 2x +1}{\sqrt{x}} dx$$ First term: $x^3 = \frac{1}{4}x^4$ Second term: $2x = x^2$ Third term: $1 = x$ Fourth term: $\sqrt{x} = x^{1/2}$ I know the fourth term is wrong and ...
2
votes
4answers
26 views

Indefinite Integral of a function

$$\int \left(\frac15 x^3 - 2x + \frac3x + e^x \right ) \mathrm dx$$ I came up with $$F=x^4-x^2+\frac{3x}{\frac12 x^2}+e^x$$ but that was wrong.
1
vote
2answers
123 views

Help me understand integration with restrictions

So, for example, we have the integral: $$I = \int{\sqrt{1+\sin x}}dx$$ and using the WA method, we solve it like this: Can you please explain me the last step, why does the solution change due to ...
0
votes
4answers
88 views

Integrate $\int\frac{dx}{(x^2+16)^3}$

Solve the following integral: $$\int\frac{dx}{(x^2+16)^3}$$ I have no idea what to do here. I think there will be trigonometric substitution but I can't even seem to get started with this problem. ...
0
votes
1answer
16 views

How did they use the chain rule to seperate the variables ?

"You have 1/x dx/dy = 1/y, and you're assuming that x is a function of y. From this you can conclude that an antiderivative of the left hand side of the above differs from an antiderivative of the ...
1
vote
1answer
22 views

Solve an Integral one open and one complex limit

If we pose an integral $2ie^{-\zeta^2}\int\limits_{-\infty}^{i\zeta} e^{-x^2} dx$ where $\zeta$ is a complex number. For imaginary $\zeta$ this comes down to the error function. I would appreciate ...
0
votes
0answers
26 views

How does this integral change due to restricted x values?

So, I integrated this integral: $$I = \int\frac{dx}{(x^2+4)\sqrt{1-x^2}}$$ And got: $$I = \frac{1}{2\sqrt{5}}\arctan{\frac{x\sqrt{5}}{2\sqrt{1-x^2}}}+C$$ But, since $-1 < x < 1$, how does it ...
2
votes
3answers
69 views

Help with integrating $\int\frac{3x}{\sqrt x} dx ?$ [closed]

How do I integrate $ \int\dfrac{3x}{\sqrt x} dx ?$ Thanks for any help
2
votes
8answers
94 views

Integrating $\int \frac{dx}{1+\frac{a}{x}}$

First-year problem: How do I integrate $$ \int \frac{dx}{1+\frac{a}{x}}? $$ My guess is $u$-substitution, or partial fractions, but nothing that I try seems to work...
0
votes
2answers
49 views

Finding the integral of the type $\frac{px+q}{ax^2 +bx + c}$

The textbook says, to find the integral of the type $\dfrac{px+q}{ax^2 +bx + c}$, where $p,q,a,b,c$ are constants, we are to find real numbers $A$ and $B$ such that $$px+q = A \dfrac{d}{dx} (ax^2 + ...
1
vote
1answer
60 views

Can someone show me if my steps are wrong in solving $\int \frac{\sqrt{1+\ln x}}{x \ln x} dx$

I just need to know if this method of solving this integral $$ \int \frac{\sqrt{1+\ln x}}{x \ln x} dx $$ is correct or not and if not where am I wrong?
1
vote
2answers
32 views

Integral of $\sin(ax+b)\cos(ax+b)$

$\displaystyle\int \sin(ax+b)\cos(ax+b) \, dx$ is giving me two answers; $\dfrac{1}{2a}\sin^2(ax+b)+C$ negative of $\dfrac{1}{4a}\cos2(ax+b) +C$ Are these two results the same? Can you please ...
0
votes
4answers
67 views

Integral of $\ln(\cos x) \tan(x)$

How can I find $$\int \ln(\cos x)\tan(x) dx$$ I'm having a hard time all the site I usually use to check doesn't work for this for some reason... I got $$\frac{\ln(\cos x)}{2}$$
3
votes
2answers
50 views

Find the following indefinite integral: $\int (x^2+6x+5)^{12} (x+3) \ dx$

The solution I got was $(1/13)(x^2+6x+5)^{13} + C$ I am not sure if I am correct though and would like help. Thanks!
0
votes
4answers
35 views

Questions on Indefinite Integration [closed]

Hi would like to know who yall will go about integrating this function. I have tried by substituting with $u=2(x-1)^{0.5}$ for question 1
0
votes
2answers
37 views

Evaluate the indefinite integral $\int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $ (using substitution) The answer is: $\frac {1}{24} ...