Questions tagged [indefinite-integrals]
Question about finding the primitives of a given function, whether or not elementary.
5,898 questions
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4
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Dummy variable rule for indefinite integrals?
My friend is tutoring high school mathematics, and one of the techniques taught is to let an integral be $I$ then get $I = abc - I$ so that $I = abc/2.$ For example, $$
I := \int e^x\cos{x} dx = eˣ \...
- 463
5
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1
answer
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The Multiplicative Role of $dx$ in Indefinite and Definite Integrals: A Comparison with Derivative Notation
I understand from prior discussions (e.g., What does the $dx$ mean in the notation for the indefinite integral?) that $dx$ in $\int f(x) \, dx$ serves as more than mere notation for the variable of ...
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Question about integral of a multivariable function
How should I simplify this expression?
$$g'(t)\cdot \int f(x)\,dx$$
Where $t$ is a constant relative to $x$.
I have a few ideas for what it might be, but I’m new to integrals of functions with ...
- 382
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1
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Evaluate $\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \operatorname{li}(x) \cos(\ln x)]}{x \ln x} \, \mathrm {dx}$
Evaluate: $$\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \operatorname{li}(x) \cos(\ln x)]}{x \ln x} \, \mathrm {dx}$$
My approach:
$$\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \...
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votes
3
answers
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How can I evaluate $\int \frac{\left(x+2\right)^{\frac{1}{6}}}{\sqrt{\left(x+2\right)^{\frac{1}{3}}+1}}\:dx$
I want to calculate
$$\int \frac{\left(x+2\right)^{\frac{1}{6}}}{\sqrt{\left(x+2\right)^{\frac{1}{3}}+1}}\:dx$$
I used $t = (x+2)^{1/3}$.
Then $x = t^3 - 2$ and $dx = 3t^2\,dt$.
Also $(x+2)^{1/6} = t^{...
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Help with simplifying this integral
Is it possible to convert this expression:
$$\int u^2 \, f''(u) \, du$$
Into some integral of this form:
$$\int t^n \, f^{(n+1)}(t) \, dt$$
Using multiple integration techniques repeatedly like ...
- 382
2
votes
2
answers
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Integral $\int_0^\infty e^{-a^2x-\frac{b^2}{x}}x^{-\frac{1}{2}} dx$ for $a>0, b\geq0$
I came across the following integral $$I=\int_0^\infty e^{-a^2x-\frac{b^2}{x}}x^{-\frac{1}{2}}dx$$ for real parameters $a>0$ and $b\geq0$. My notes say that the solution is $$I=\frac{\sqrt{\pi}}{a}...
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Evaluate $\int[\tan^{4}(x)\sec^{3}(x)+\tan^{2}(x)\sec^{5}(x)]dx$
I need help in evaluating
$$\int\left[\tan^{4}(x)\sec^{3}(x)+\tan^{2}(x)\sec^{5}(x)\right]dx$$
This Integral is from the MIT Integration BEE 2022
Let's Assume $$I=\int\left[\tan^{4}(x)\sec^{3}(x)+\tan^...
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$\int \frac{e^{x^2}}{\sqrt{1+x^2}} \, dx$
I'm working on the following indefinite integral and am struggling to find an elegant solution. I've tried some standard substitution methods, but I can't seem to simplify it into something more ...
- 43
2
votes
5
answers
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Evaluate $\int^{\infty}_{1}\frac{1}{(x+1)\sqrt{x^2+2x-2}}\,\mathrm dx$
I found this integral in STEP:
$$\int^{\infty}_{1}\frac{1}{(x+1)\sqrt{x^2+2x-2}}\,\mathrm dx$$
My approach:
When I saw $x^2+2x-2$, I try to complete the square,
$$\int\frac{1}{(x+1)\sqrt{x^2+2x-2}}\,\...
- 491
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1
answer
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How to take the integral $\int \frac{\sqrt{\sin(x)}}{(1+\sin^2x)}dx$?
$$\int \frac{\sqrt{\sin(x)}}{(1+\sin^2x)}dx$$I found this integral in a well-known problem book. It states that the integral has a solution and provides an answer. I have tried many approaches to ...
- 21
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Integrating $x \, f’(x)$, where $f(x)$ is known
I am working on a method of integration that involves having to work out the integral of:
$$x \,f’(x)$$
which is the integral of $x$ times the derivative of $f(x)$ (where $f(x)$ is known).
Obviously, ...
- 382
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votes
6
answers
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Integral $\int\frac{x^2}{x^4+2x^2+2}\mathrm dx$
I'm trying to solve the following integral:
$$\int\frac{x^2}{x^4+2x^2+2}\mathrm dx.$$
I've been trying various methods to solve this, but I'm completely stuck and could use some guidance. I'm looking ...
- 81
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1
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Derivation of the integrating factor
Is this a valid derivation of the integrating factor?
\begin{align}&\frac{d\mu }{dx}=\mu P(x) \longrightarrow \frac{1}{\mu }d\mu =P(x)dx\longrightarrow \int \frac{1}{\mu }d\mu =\int P(x)dx\\\...
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Closed form formula for $\int \frac{dx}{x(\sin x+1)}$
I am trying to solve the following differential equation:
$$
x \frac{dy}{dx} = y \sin\left(\frac{y}{x}\right) + 2y.
$$
Using the substitution $v = \frac{y}{x}$, we have $y = vx$ and
$\frac{dy}{dx} = v ...
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