Questions tagged [indefinite-integrals]
Question about finding the primitives of a given function, whether or not elementary.
5,903 questions
3 votes
2 answers
156 views
Alternative methods for $\int \sqrt{\frac{1 - \sqrt{x^2 + 1}}{x}} \, dx$
Consider the integral $$I=\int \sqrt{\frac{1 - \sqrt{x^2 + 1}}{x}} \, dx\qquad(\text{for}\,x<0) $$ My solution is based on Transform 3 and 5 in this draft. $$\begin{aligned}I &= \int \sqrt{\...
- 1,223
2 votes
1 answer
84 views
How are these integral behaviour approximations different
Q1. $$\int \frac{3x^{2} + 4x - 1}{(x^{2} + 1)^{2}\sqrt{x+1}}\, dx$$ $\textbf{A. }\frac{\sqrt{x+1}}{x^{2}+1} + C$ $\textbf{B. }-\frac{2\sqrt{x+1}}{x^{2}+1} + C$ $\textbf{C. }-\frac{x}{(x^{2}+1)\sqrt{x+...
-3 votes
0 answers
76 views
Is this integral possible to solve without special functions? [closed]
$$\int \frac{3x^2+4}{2\sqrt{x}(4-3x^2)\sqrt{3x^2+x-4}}\ dx$$
-4 votes
0 answers
60 views
Meaning of the integral sign without any variable of integration [closed]
Recently, I’ve been learning about doing algebra with$\def\d{\mathrm d}$ $\d x$ (resp., $\d y$, $\d t$, $\d f$), and I know it’s essentially an infinitesimal difference in $x$ (resp., $y$, $t$, $f(x)$)...
- 9
2 votes
1 answer
114 views
How to evaluate $ \int \frac{x}{a_2 x^2 + a_1 x + a_0 + \sqrt{b_4 x^4 + \ldots + b_0}} \, dx $?
I want to evaluate $$ \int \frac{x}{a_2 x^2 + a_1 x + a_0 + \sqrt{b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + b_0}} \, dx $$ where $a_i , b_i \in \mathbb{R}$ and generally nonzero. This integral arises as a ...
- 311
4 votes
1 answer
261 views
On $\int_{0}^{\frac{\pi}{2}} \sin(x)^{-\sin(x)}$, and other similar "Sophomore's dream"-esque integrals.
This is a problem I have been tussling with for a while, with varying progress through the years. I think I am finally stuck, or at least cannot move forward with knowing significantly more. As such, ...
- 41
8 votes
4 answers
824 views
Dummy variable rule for indefinite integrals?
My friend is tutoring high school mathematics, and one of the techniques taught is to let an integral be $I$ then get $I = abc - I$ so that $I = abc/2.$ For example, $$ I := \int e^x\cos{x} dx = eˣ \...
- 473
5 votes
1 answer
345 views
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 ...
- 321
0 votes
0 answers
73 views
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 ...
- 424
1 vote
1 answer
121 views
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: $$\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \...
- 491
15 votes
3 answers
137 views
How can I evaluate $\int \frac{\left(x+2\right)^{\frac{1}{6}}}{\sqrt{\left(x+2\right)^{\frac{1}{3}}+1}}\:dx$
I want to calculate $$\int \frac{\left(x+2\right)^{\frac{1}{6}}}{\sqrt{\left(x+2\right)^{\frac{1}{3}}+1}}\:dx$$ I used $t = (x+2)^{1/3}$. Then $x = t^3 - 2$ and $dx = 3t^2\,dt$. Also $(x+2)^{1/6} = t^{...
- 151
0 votes
0 answers
161 views
Help with simplifying this integral
Is it possible to convert this expression: $$\int u^2 \, f''(u) \, du$$ Into some integral of this form: $$\int t^n \, f^{(n+1)}(t) \, dt$$ Using multiple integration techniques repeatedly like ...
- 424
2 votes
2 answers
162 views
Integral $\int_0^\infty e^{-a^2x-\frac{b^2}{x}}x^{-\frac{1}{2}} dx$ for $a>0, b\geq0$
I came across the following integral $$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}...
- 23
2 votes
3 answers
122 views
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^...
- 1,836
1 vote
1 answer
157 views
$\int \frac{e^{x^2}}{\sqrt{1+x^2}} \, dx$
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