# Questions tagged [indefinite-integrals]

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

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### How to get to the correct result for this integral?

Wolfram|Alpha is, as far as I know, the only website that gives the correct solution to this integral, $$\int \sqrt{2+\sqrt{2+\sqrt{2+2\cos(5\sqrt x+4)}}} \, x^{-1/2}\, dx$$ because deriving the ...
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### Indefinite integral $\int \frac{\log\Gamma(\frac{x}{2})}{x^2}\,dx$ [closed]

I want to compute the indefinite integral $$\int \frac{\log\Gamma(\frac{x}{2})}{x^2}\,dx,$$ but fail myself. Do you know how to compute this integral?
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### Can someone please explain this integral property of odd functions [duplicate]

I came across this integral in a text $$\int_{-1}^{1} \frac{cosx}{e^{\frac{1}{x}}+1}.$$ This was the approach used in the text: Let $$g(x)= \int_{-1}^{1} \frac{\cos x}{e^{\frac{1}{x}}+1}$$ Then $g(x)$ ...
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$$\Omega = \int_{-1}^1 \frac{dx}{1+x²}\ =\ -\int_{-1}^1\frac{dy}{1+y²}$$ $$\text{ by using the transformation }\; x= \frac{1}{y}$$ $$\Omega = - \Omega ,\ 0\,\text{hence } \Omega =0$$ but the ...
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### Evaluate $\int\frac{\mathrm{d}x}{{(x^4+2x+10)}^4}$

I am having trouble with this integral $\displaystyle \int \frac{\mathrm{d}x}{{(x^4+2x+10)}^4}$ I know this can be solved using the "ostrogradsky" method. But that is too lengthy. i tried ...
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### How do I understand if an indefinite integral is not solvable in terms of elementary functions? [duplicate]

Many indefinite integrals cant be solved in terms of elementary functions.e.g.-$$\int \frac{\sin(x)}{x}dx$$ But many hard looking ones are still solvable by weird substitutions and other tricks. Is ...
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### Evaluate $\int{(\sin x + 2\cos x)}^3 \, dx$

I tried to expand the binomial. But its complicated. $\int{\sin x}^3 + {(2\cos x)}^3 + 3\sin x2\cos x (\sin x + 2\cos x)\, dx$ I dont think i can use integral by parts on it. Using substitution ...
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### General formula for integral of polynomial expressed as roots product

I’m wondering whether there is a general formula to integrate a polynomial expressed as product of its $n$ roots, in other words $\int p(x)dx$ With $p(x) = (x-r_1) (x-r_2)...(x-r_n)$ So to have the ...
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### Extending Feynman's integral trick to indefinite integrals

Question The Feynman's integral trick (a.k.a. differentiation under the integral) proves to be a powerful tool when dealing with definite integrals. However, I have been unable to find a similar ...
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### What's wrong with this method of evaluating an integral?

I was trying to evalute the integral $$\int \frac{1}{x^2+1} \,dx$$ by partial fractions. $$\frac{1}{x^2+1} = \frac{1}{2i}\left(\frac{1}{x-i} - \frac{1}{x+i}\right)$$ Therefore, \begin{aligned} \int \...
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### How to Integrate $\sqrt{a+b\tan x}$? [closed]

How can I evaluate the integral below? $$\int\sqrt{a+b\tan x}\ dx$$
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### Integrating $\int \sqrt{a^2+x^2} \ \mathrm{d} x$ with trig. substitution

I am trying to come up with all the formulas I have myself and I stumbled upon a roadblock again. Integrating $\int \sqrt{a^2+x^2} \ \mathrm{d} x$ with Trig Substitution. So I imagined a triangle ...
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### Indefinite integral of $\sin^8(x)$

Suppose we have the following function: $$\sin^8(x)$$ We have to find its anti-derivative To find the indefinite integral of $\sin^4(x)$, I converted everything to $\cos(2x)$ and $\cos(4x)$ and then ...
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### Integrate $\frac{x}{(x^4+1)^2 \sqrt{(x^2-x+1)}}$. [closed]

Evaluate the indefinite integral, $$\int\frac{x}{(x^4+1)^2 \sqrt{(x^2-x+1)}} \mathrm{d}x$$ Found this problem in a mathematics group site, but the solution was never posted. I suspect it cannot be ...
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### What is the problem with this method while integrating $(e^x-(2x+3)^4)^3$?

What is the problem with this method while integrating $(e^x-(2x+3)^4)^3$? I already know what is its integration. I collected the answers from Quora (black ones) and WolframAlpha website (the red one)...
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### $\int ^{\infty}_0 \frac{\sin(x)}{x}dx=\frac{\pi}{2}$ using only path substitution of $\frac{e^{iz{}}}{z}$ [duplicate]

Let's define path substitution as: $\int f=\int _\gamma \gamma'(t) f(\gamma(t)) dt$ where $\gamma:\mathbb{R}\rightarrow \mathbb{R}$. Use only path substitution on $f(z)=\dfrac{e^{iz{}}}{z}$ to prove ...
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### Mistake in evaluating $\int {\frac{\cos 2x}{(\cos x+\sin x)^2 }}dx$

I know I've got the incorrect answer. Can anyone spot where I did wrong?
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### How to evaluate/simplify integration with 4 parts?

How might I evaluate the following indefinite integral? $$\int k \, x^a \,(1-x)^b \,(x-y)^c \, (1-x+y)^d \,\,dx$$ The aim is to get a function of $y$ once I put in my limits, but $k$, $a$, $b$, $c$, ...
### How to evaluate $\int_{0}^{\infty} x^{\nu} \frac{e^{-\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}} \, dx$?
$$\int_{0}^{\infty} x^{\nu} \frac{e^{-\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}} \, dx$$ Is it possible to calculate this for $a>0$ and $\nu=0, 2$ ? I think the result seems to include exponential integral ...