Questions tagged [indefinite-integrals]

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

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3
votes
1answer
61 views

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 ...
-1
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1answer
30 views

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?
0
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1answer
44 views

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 $g(x)$ ...
4
votes
2answers
85 views

Can someone please help me explain this fallacy [closed]

$$\Omega = \int_{-1}^1 \frac{dx}{1+x²}\ =\ -\int_{-1}^1\frac{dy}{1+y²} $$ $$ \text{ by using the transformation }\; x= \frac{1}{y}$$ $$ \Omega = - \Omega ,\ 0\,\text{hence } \Omega =0 $$ but the ...
2
votes
1answer
98 views

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 ...
0
votes
0answers
33 views

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 ...
0
votes
5answers
75 views

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 ...
3
votes
2answers
60 views

Integration of $\sqrt {\tan x}$ [duplicate]

I have tried many ways to integrate $\sqrt {\tan x}$ including integration by parts but didn't get to any final result. I also assumed, $$ \tan x = t^2 $$ $$ \int \sqrt {\tan x} \,dx $$ $$⇒\int \frac{...
4
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0answers
65 views

How to apply Risch's algorithm to $\int\frac{x}{\sqrt{x^4-2x^3+3x^2+4x+1}}\,\mathrm{d}x$?

It is known that $$\int\frac{x}{\sqrt{x^4-2x^3+3x^2+4x+1}}\,\mathrm{d}x=$$ $$-\frac{1}{6}\log\Big((2x^4-10x^3+24x^2-28x+14)\sqrt{x^4-2x^3+3x^2+4x+1}-2x^6+12x^5-36x^4+56x^3-42x^2+13\Big)+C.$$ To get ...
2
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0answers
95 views

Why does the Fundamental Theorem of Calculus need to be applied to derive the power rule for integration?

On Wikipedia, the article about the power rule states that if $f:\mathbb{R}\to\mathbb{R}$ is given by $f(x)=x^n$, then $$ f'(x)=nx^{n-1} $$ provided that $f$ is differentiable at $x$, $n$ is a ...
0
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0answers
31 views

Relating $u$-substitution in indefinite integrals to a definite integral.

I've always thought of dx, the variable of integration as the thing that tells us what variable we are integrating with respect to, in indefinite integration, and as the vanishingly small width of a ...
3
votes
2answers
225 views

Prove: $\int_0^1 \int_0^1 \frac{\ln{\left(2+yx\right)}}{1+yx} \; \mathrm{d}y\; \mathrm{d}x = \frac{13}{24} \zeta(3)$

Prove: $$\int_0^1 \int_0^1 \frac{\ln{\left(2+yx\right)}}{1+yx} \; \mathrm{d}y\; \mathrm{d}x = \frac{13}{24} \zeta(3)$$ My attempt: \begin{align} \int_0^1 \int_0^1 \frac{\ln{\left(1+(1+yx)\right)}}{1+...
2
votes
2answers
120 views

Ideas for this integral: $\int \frac{\sqrt{\tan{x}}}{\sin{x}} dx$

$$\int \frac{\sqrt{\tan{x}}}{\sin{x}} \mathrm{d}x$$ So I was wondering if this correctly by converting $\sqrt{\tan{x}}$ into $\frac{\sqrt{\sin{x}}}{\sqrt{\cos{x}}}$ therefore I can divide it with $\...
3
votes
1answer
61 views

Indefinite integral of $\frac{\sec^2x}{(\sec x+\tan x)^\frac{9}{2}}$

$$\frac{\sec^2x}{(\sec x+\tan x)^\frac{9}{2}}$$ My approach: Since it is easy to evaluate $\int{\sec^2x}$ , integration by parts seems like a viable option. Let $$I_n=\int{\frac{\sec^2x}{(\sec x+\tan ...
0
votes
1answer
98 views

Integral of $\int\limits_0^{2\pi } {\operatorname{erfc}\left( {\cos \left( {a + \theta } \right)} \right)d\theta } $?

I am sorry if it does not fit here. I found some of the integral for the complementary error function e.g. So far I did not find any integral regarding, $\int\limits_0^{2\pi } {\operatorname{erfc}\...
2
votes
2answers
109 views

Why must $\int_\gamma f(z)\;d z = 0$ for *any* contour $γ$ to define antiderivative of $f$?

Whilst I was reading the following proposition from Dexter Chua's lecture notes on Complex Analysis: Let $U \subseteq \mathbb{C}$ be a domain (i.e. path-connected non-empty open set), and $f: U \to \...
0
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0answers
21 views

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 ...
1
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0answers
81 views

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 ...
0
votes
2answers
108 views

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 \...
1
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0answers
88 views

How to Integrate $\sqrt{a+b\tan x}$? [closed]

How can I evaluate the integral below? $$\int\sqrt{a+b\tan x}\ dx$$
1
vote
1answer
53 views

Evaluating $\int \left(\frac{1}{3x}-2\sec^2\left(\frac x2\right)-e^{-2x+3}\right)dx$

Given that $\displaystyle f(x)=\frac{1}{3x}-2\sec^2\left(\frac{x}{2}\right)-e^{-2x+3}$, evaluate $\int f(x) \mathrm{d}x$. Attempt: $$\begin{aligned} \int f(x) \mathrm{d}x&=\int \left[ \frac{x}{3}-...
2
votes
1answer
131 views

How do I integrate $\frac1{x^2+x+1}$?

I have tried this: $$\frac1{x^2+x+1} = \frac1{\left( (x+\frac12)^2+\frac34\right)}$$ Now $u = x+\frac12$ $$\frac1{ u^2+\frac34 }$$ Now multiply by $ \frac34$ $$\frac1{ \frac43 u^2 + 1}$$ Now put the $...
1
vote
3answers
91 views

Need help with finishing integration

I have the following integral: $$y=\int \frac{1}{1-2\sqrt{x}} \, dx$$ I first got $u=2\sqrt{x}$ which gives us $x=\frac{u^2}{4}$. Plugging this in I got: $$y=\int \frac{1}{1-2\sqrt{\frac{u^2}{4}}} \, ...
1
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3answers
101 views

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 ...
1
vote
3answers
65 views

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 ...
4
votes
3answers
139 views

Evaluate integral $\int (x^2-1)(x^3-3x)^{4/3} \mathop{dx}$

How can I evaluate this integral $$\int (x^2-1)(x^3-3x)^{4/3} \mathop{dx}=\;\;?$$ My attempt: I tried using substitution $x=\sec\theta$, $dx=\sec\theta\ \tan\theta d\theta$, $$\int (\sec^2\theta-1)(\...
4
votes
1answer
64 views

Why don't we draw modulus bars when we open 'under root' in indefinite integration?

Is there a way we can ignore absolute value bars in indefinite integration. Below is the solution of a problem that confused me and not just this problem, I recently noticed that all the problem and ...
2
votes
0answers
65 views

Evaluate the general integral $\int \sin(ax) \sin^b(x) dx $

Evaluate the indefinite integral $$I(a,b)=\int \sin(ax) \sin^b(x)\mathrm{d}x \hspace{40pt} a,b\in\mathbb{N}$$ How do we evaluate the above indefinite integral? Here is a question with $a=2015$ and $b=...
1
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0answers
38 views

Bessel functions: Negative integer order

I would like to show that for the Bessel functions, we have: $J_{-m}(x) = (-1)^m J_m(x)$. This can be obtained by using the definition of the Bessel functions as $J_m(x) = \big(\frac{x}{2}\big)^m \...
0
votes
2answers
72 views

Why $\int \frac{dx}{(1-x)^2} = \frac{1}{1-x}$ and $\int \frac{dx}{(1-x)^2} = \frac{x}{1-x}$ are both correct? [duplicate]

Why $\int \frac{dx}{(1-x)^2} = \frac{1}{1-x}$ and $\int \frac{dx}{(1-x)^2} = \frac{x}{1-x}$ are both correct? I can differentiate both $\dfrac{1}{1-x}$ and $\dfrac{x}{1-x}$ to get the integrand. But ...
1
vote
2answers
77 views

Correct way to integrate $\int x(x^2-16)dx$

Evaluate: $$\int x(x^2-16)dx$$ I have noticed that this integral can be solved using two different methods, but I am not sure which one is the correct one. Way 1: Using $u$-subtitution Let $u=x^2-16, ...
1
vote
3answers
85 views

Why does making the 'wrong' $u$-substitution still work in this example?

I was evaluating the integral $$ \int 2x \cos(x^2)dx $$ and realised that it could be written in the form $$ \int f'(g(x))g'(x)dx $$ and so substitution could be used to help evaluate it. Setting $u$ ...
0
votes
2answers
92 views

Evaluate $ \int\left(x^{3} + x^{6}\right)\left(x^{3} + 2\right)^{1/3} \,\mathrm{d} x $

Integrate : $$ \int\left(x^{3} + x^{6}\right)\left(x^{3} + 2\right)^{1/3} \,\mathrm{d} x $$ I have tried assuming that ${{x}^{3}}+2$ term as ${u}^{3}$, $dx=u^2du/(u^3-2)^{2/3}$ $$ \int\left(x^{3} + x^{...
2
votes
3answers
106 views

How to integrate $\int {2\over (x^2+2)\sqrt{x^2+4}}dx$?

Solve the following indefinite integral: $$\int \frac{2}{(x^2+2)\sqrt{x^2+4}} dx$$ My approach: I used the substitution: $x=2\tan t$, $dx=2\sec^2t dt$ $$\int \frac{2}{(x^2+2)\sqrt{x^2+4}} dx=\int \...
0
votes
1answer
43 views

About First and second Fundamental Theorem of Calculus

Define $\displaystyle f(x)=\int_{0}^{x}e^{-t^2}\mathrm{d}t$. Evaluate $\displaystyle\int f(x)~\mathrm{d}x$. The final answer may involve $f(x)$. I know about $\displaystyle f'(x)=\frac{\mathrm{d}}{\...
0
votes
2answers
21 views

Need help with integral manipulation

I have the following integral, I need to solve $$\int \frac{f'(t)}{af(t) - bf(t)^2}dt$$ where $a$ is a constant and $b \in (0,1)$. I used substitution. Let $u = f(t)$ and let $du = f'(t) dt$. I then ...
0
votes
0answers
81 views

Is this a good justification for why integration by substitution works? [duplicate]

It is a well known fact that $$ \int f'(g(x))g'(x)dx=f(g(x))+C $$ This follows directly from the chain rule. However, sometimes it is easier to perform the substitution $u=g(x)$: \begin{align} u&=...
2
votes
2answers
147 views

Solving $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{xy+{y}^{2}}{{x}^{2}+{y}^{2}}$.

Solve $$\frac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{xy+{y}^{2}}{{x}^{2}+{y}^{2}}.$$ I have tried to solve this question by assuming $y/x$ to be $v$, but I am stuck on the integral $$\int\frac{{v}^{2}+1}{{...
-1
votes
1answer
88 views

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 ...
1
vote
2answers
72 views

How to solve the ODE $y' = \frac{x+y-2}{y-x-4}$?

I am trying to solve the ODE $$y' = \frac{x+y-2}{y-x-4} \tag1 $$ This is a homogeneous special form ODE. Let $x = u -1$ and $y=v+3$ in order to transform it to a homogeneous ODE. Hence, $$ (1) \iff v'...
0
votes
1answer
37 views

How to evaluate $\int \frac{\left(2-z\left(u\right)\right)z'\left(u\right)}{\left(z\left(u\right)\right)^2-z\left(u\right)+2}du$?

I am trying to evaluate $$ \int \frac{\left(2-z\left(u\right)\right)z'\left(u\right)}{\left(z\left(u\right)\right)^2-z\left(u\right)+2}du \quad (1) $$ I think I've found a solution by using ...
7
votes
4answers
127 views

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

Evaluate: $$\int \frac{2-x^3}{(1+x^3)^{3/2}} dx$$ I could find the integral by setting it equal to $$\frac{ax+b}{(1+x^3)^{1/2}}$$ and differentiating both sides w.r.t.$x$ as $$\frac{2-x^3}{(1+x^3)^{3/...
1
vote
1answer
80 views

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)...
-1
votes
0answers
49 views

$\int ^{\infty}_0 \frac{\sin(x)}{x}dx=\frac{\pi}{2}$ 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 ...
0
votes
2answers
56 views

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?
0
votes
0answers
33 views

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 $k$, $a$, $b$, $c$, ...
2
votes
1answer
60 views

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 ...
2
votes
0answers
41 views

Find $\int_0^{\frac{\pi}{2}} e^{-a(\sin(x)+\cos(x))} \, dx$ and/or $\int_0^{\frac{\pi}{2}} e^{-a(\sin(x)+\cos(x))} \sin(\cos(x)) \, dx$

While trying to solve a certain integral I was left with these rather silly integral. $$\int_0^{\frac{\pi}{2}} e^{-a(\sin(x)+\cos(x))} \, dx$$ $$\int_0^{\frac{\pi}{2}} e^{-a(\sin(x)+\cos(x))} \sin(\...
2
votes
5answers
176 views

Other way to evaluate $\int \frac{1}{\cos 2x+3}\ dx$?

I am evaluating $$\int \frac{1}{\cos 2x+3} dx \quad (1)$$ Using Weierstrass substitution: $$ (1)=\int \frac{1}{\frac{1-v^2}{1+v^2}+3}\cdot \frac{2}{1+v^2}dv =\int \frac{1}{v^2+2}dv \quad (2) $$ And ...
1
vote
4answers
130 views

How to evaluate $\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} dx$?

I am trying to evaluate $$\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} dx \quad (1)$$ The typical way to confront this kind of integrals are the conjugates i.e: $$\int \frac{\sqrt{1+x}+\...

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