Questions tagged [indefinite-integrals]

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

212
votes
15answers
27k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
147
votes
19answers
23k views

Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your ...
106
votes
9answers
5k views

The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)

In general, the indefinite integral of $x^n$ has power $n+1$. This is the standard power rule. Why does it "break" for $n=-1$? In other words, the derivative rule $$\frac{d}{dx} x^{n} = nx^{n-1}$$ ...
93
votes
11answers
8k views

Ways to evaluate $\int \sec \theta \, \mathrm d \theta$

The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then ...
85
votes
11answers
16k views

Demystify integration of $\int \frac{1}{x} \mathrm dx$

I've learned in my analysis class, that $$ \int \frac{1}{x} \mathrm dx = \ln(x). $$ I can live with that, and it's what I use when solving equations like that. But how can I solve this, without ...
79
votes
8answers
8k views

Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$

I'm having trouble computing the integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$ I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as ...
72
votes
4answers
3k views

Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions

Let $f_n(x)$ be recursively defined as $$f_0(x)=1,\ \ \ f_{n+1}(x)=\sqrt{x+f_n(x)},\tag1$$ i.e. $f_n(x)$ contains $n$ radicals and $n$ occurences of $x$: $$f_1(x)=\sqrt{x+1},\ \ \ f_2(x)=\sqrt{x+\sqrt{...
49
votes
6answers
6k views

How can this function have two different antiderivatives?

I'm currently operating with the following integral: $$\int\frac{u'(t)}{(1-u(t))^2} dt$$ But I notice that $$\frac{d}{dt} \frac{u(t)}{1-u(t)} = \frac{u'(t)}{(1-u(t))^2}$$ and $$\frac{d}{dt} \frac{...
47
votes
6answers
45k views

Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $

I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$
32
votes
2answers
1k views

Evaluate $\int\frac1{1+x^n}dx$ for $n\in\mathbb R$

I was wondering on how to evaluate the following indefinite integral for all $n\in\mathbb R$. $$\int\frac1{1+x^n}dx$$ It seems to be peculiar in that we have $$\begin{align} \int\frac1{1+x^{-1}}dx&...
27
votes
2answers
8k views

When the integral of products is the product of integrals.

I'm self-studying and was doing the following integral: $$I = \int \frac{e^{\frac{1}{x}+\tan^{-1}x}}{x^2+x^4} dx $$ I solved it fine by letting $ u = \frac{1}{x} + \tan^{-1}x$. My question is ...
26
votes
1answer
1k views

Proof that $\int x^x dx$ can't be done in terms of elementary functions?

Is there any easy proof that $\int x^x dx$ can't be done in terms of elementary functions? I know this is true, because the Risch algorithm gives a decision process for integration in terms of ...
25
votes
2answers
2k views

Is indefinite integration non-linear?

Let us consider this small problem: $$ \int0\;dx = 0\cdot\int1\;dx = 0\cdot(x+c) = 0 \tag1 $$ $$ \frac{dc}{dx} = 0 \qquad\iff\qquad \int 0\;dx = c, \qquad\forall c\in\mathbb{R} \tag2 $$ These are two ...
25
votes
0answers
1k views

How can I calculate $\int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$ [duplicate]

How can I calculate $$ \int{\sec\left(x\right)\tan\left(x\right) \over 3x + 5}\,{\rm d}x $$ My Try:: $\displaystyle \int \frac{1}{3x+5}\left(\sec x\tan x \right)\,\mathrm dx$ Now Using Integration ...
24
votes
1answer
617 views

Evaluate $\int (1-x^{2008})^{\frac{1}{2007}} (1-x^{2007})^{\frac{1}{2008}} dx$

Evaluate the given integral $$\int (1-x^{2008})^{\frac{1}{2007}} (1-x^{2007})^{\frac{1}{2008}} dx$$ Using integration by parts is out of equation because we can't integrate any of two brackets. I ...
22
votes
3answers
1k views

How to find $\int x^{1/x}\mathrm dx$

EDIT: The full answer has been posted by myself. Feel free to check the logic within. How does one indefinitely integrate a function in the form of $$f(x)=x^{1/x}$$ Looking at all the things that I ...
22
votes
2answers
457 views

When is ${\large\int}\frac{dx}{\left(1+x^a\right)^a}$ an elementary function?

Consider the following indefinite integral: $$F_a(x)=\int\frac{dx}{\left(1+x^a\right)^a}.$$ Here $a\in\mathbb R$ is a parameter, and $x>0$ is a variable. For what values of the parameter $a$ the ...
22
votes
1answer
1k views

Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $

Background: On my quest to solve difficult integrals, I chanced upon this site: http://www.durofy.com/5-most-beautiful-questions-from-integral-calculus/ Good problems for me, (novice), although I ...
21
votes
6answers
2k views

Why isn't $\int \frac{1}{x}~dx = \frac{x^0}{0}$?

I know that $\int \frac{1}{x}~dx = \ln|x| + C$ and I know the antiderivative method works for all powers of $x$ except $-1$. But why is that the case? I am still in high school and teachers aren't ...
21
votes
3answers
907 views

Evaluation of $\int\frac{5x^3+3x-1}{(x^3+3x+1)^3}\,dx$

Evaluate the integral $$\int\frac{5x^3+3x-1}{(x^3+3x+1)^3}\,dx$$ My Attempt: Let $f(x) = \frac{ax+b}{(x^3+3x+1)^2}.$ Now differentiate both side with respect to $x$, and we get $$ \begin{align} f'...
21
votes
3answers
4k views

$-1 = 0$ by integration by parts of $\tan(x)$

I had a calculus final yesterday, and in a question we had to find a primitive of $\tan(x)$ in order to solve a differential equation. A friend of mine forgot that such a primitive could easily be ...
21
votes
4answers
527 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+1/3 }{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted $x=\dfrac{1}{...
21
votes
3answers
658 views

Evaluating $\int{ \frac{\arctan\sqrt{x^{2}-1}}{\sqrt{x^{2}+x}}} \,dx$

How to integrate? $$\int{ \frac{\arctan\sqrt{x^{2}-1}}{\sqrt{x^{2}+x}}}\, dx$$ I have no idea how to do it. Tried to get some information from wiki, but its too hard :|
19
votes
5answers
99k views

Is there a rule of integration that corresponds to the quotient rule?

When teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivatives ...
19
votes
4answers
394 views

Is there no formula for $\cos(x^2)$?

I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance $$\...
19
votes
3answers
719 views

Are there any special rules when making a substitution in an integral?

Please consider the integral: $$\int_{-a}^{a}x^2dx=\frac{2a^3}{3}$$ I would like to know why I can't make the substitution: $$u=x^2$$ When I make the substitution, the limits of the integral will ...
19
votes
2answers
578 views

Integral $\int \frac{\mathrm{d}x}{\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}}$

$$\int \frac{\mathrm{d}x}{\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}}$$ 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.
18
votes
10answers
2k views

What is the most efficient method to evaluate this indefinite integral?

$$\int x^5 e^x\,\mathrm{d}x$$ Is there another, more efficient way to solve this integral that is not integration by parts?
18
votes
9answers
2k views

Indefinite integral of secant cubed

I need to calculate the following indefinite integral: $$I=\int \frac{1}{\cos^3(x)}dx$$ I know what the result is (from Mathematica): $$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$ but I don't ...
18
votes
5answers
512 views

Evaluating this integral $ \small\int \frac {x^2 dx} {(x\sin x+\cos x)^2} $

The question: Compute$$ \int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2} $$ Tried integration by parts. That didn't work. How do I proceed?
18
votes
1answer
287 views

Is it possible to find indefinite integral of $\int \frac{1} {{\sin(x)+\sec^2(x)}}\mathrm{d}x$?

I am in standard $XI$ (i.e.11) and newbie in learning topic of integration. My friend asked me to find indefinite integral of the example shown below $$y=\int \frac{1} {{\sin(x)+\sec^2(x)}} \, \...
17
votes
5answers
2k views

Integration of secant

$$\begin{align} \int \sec x \, dx &= \int \cos x \left( \frac{1}{\cos^2x} \right) \, dx \\ &= \int \cos x \left( \frac{1}{1-\sin^2x} \right) \, dx \\ & = \int\cos x\cdot\frac{1}{1-\...
17
votes
4answers
1k views

How to simplify $\int{\sqrt[4]{1-8{{x}^{2}}+8{{x}^{4}}-4x\sqrt{{{x}^{2}}-1}+8{{x}^{3}}\sqrt{{{x}^{2}}-1}}dx}$?

I have been asked about the following integral: $$\int{\sqrt[4]{1-8{{x}^{2}}+8{{x}^{4}}-4x\sqrt{{{x}^{2}}-1}+8{{x}^{3}}\sqrt{{{x}^{2}}-1}}dx}$$ I think this is a joke of bad taste. I have tried every ...
17
votes
1answer
468 views

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion

(Note: I apreciate very much who marked this as a duplicate but I would like an answer for why my proof is wrong) This is my solution, I have no clue why it failed. Let's start: define $$I_n(m) = \...
17
votes
3answers
5k views

Indefinite double integral

In calculus we've been introduced first with indefinite integral, then with the definite one. Then we've been introduced with the concept of double (definite) integral and multiple (definite) integral....
16
votes
2answers
23k views

Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$

I am trying to prove $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ This problem is a classic, but I seem to be missing one step or the understanding of ...
16
votes
1answer
2k views

Indefinite Integral $\int\sqrt[3]{\tan(x)}dx$

For calculating $\int\sqrt{\tan(x)}dx$, I used this easy method $$\begin{align}\int\sqrt{\tan(x)}dx&=\frac{1}{2}\int\left(\sqrt{\tan(x)}+\sqrt{\cot(x)}\right)dx+\frac{1}{2}\int\left(\sqrt{\tan(x)}-...
16
votes
1answer
703 views

How do experts mentally classify indefinite integrals?

Integration is as much an art as a science. Someone who is an expert looks at an indefinite integral and classifies it in a different way than someone who is a beginner. E.g., the beginner may just ...
16
votes
2answers
360 views

Integration by parts, the cases when it does not matter what $u$ and $dv$ we choose.

I was reviewing Integration By Parts on Brilliant.org where an example they use is $$\int x \ln x \;dx$$ Let $u=\ln x$ and $dv=x\;dx$ such that $$\begin{align} \int x \ln x\;dx&\;=\;\frac ...
15
votes
3answers
1k views

What is wrong with this integral reasoning?

$$\int\frac{x^{2}+1}{x\sqrt{x^{4}+1}}dx$$ We start by multiplying by $1=\frac{x}{x}$. $$\int\frac{x^{2}+1}{x^{2}\sqrt{x^{4}+1}}xdx$$ Next, we use the substitution $u=x^{2}$;$\frac{du}{2}=xdx$. $$\frac{...
15
votes
4answers
1k views

Evaluating $\int \frac {\sqrt{\tan \theta}} {\sin 2\theta} \ d \theta$

I am trying to evaluate $$\int \frac {\sqrt{\tan \theta}} {\sin 2\theta} \ d \theta$$ I tried rewriting it as $$\int {\sqrt{\tan \theta}} \cdot \csc(2\theta) \ d\theta$$ Supposedly letting $u = \...
15
votes
1answer
744 views

Why doesn't this operation work?

In my maths class we are learning about indefinite integrals, this is the problem we were working on: $$ \int \frac{1}{2x+1}dx $$ Using u-substitution we obtain: $$ \frac{1}{2}\ln\left | 2x+1 \...
15
votes
3answers
8k views

What's the connection between the indefinite integral and the definite integral?

I want to understand the connection between the primitive function or antiderivative and the definite integral. My problem with this is the independent variable called t in the formula for the first ...
15
votes
2answers
2k views

Families of functions closed under integration

What are some concrete families $\mathcal F$ of real functions that are closed under integration in the sense that for every $f \in \mathcal F$ there is $F \in \mathcal F$ such that $F'=f$? Here are ...
14
votes
5answers
3k views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}\,dx$

Compute the indefinite integral $$ \int\frac{\sqrt{\cos 2x}}{\sin x}\,dx $$ My Attempt: $$ \begin{align} \int\frac{\sqrt{\cos 2x}}{\sin x}\,dx &= \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin ...
14
votes
4answers
3k views

Integrate : $\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}dx$

$$\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}\ dx$$ My approach : Dividing the denominator by $\cos^2x$ we get $\dfrac{x^2\sec^2x }{(x -\tan x)(x\tan x +1)}$ then $$\int \frac{x^2\sec^2x}{...
14
votes
5answers
581 views

Evaluate $\int \frac{\sec^{2}{x}}{(\sec{x} + \tan{x})^{5/2}} \ \mathrm dx$

I am unable to solve the following integral: $$\int \frac{\sec^{2}{x}}{(\sec{x} + \tan{x})^{5/2}} \ \mathrm dx.$$ Tried putting $t=\tan{x}$ so that numerator has $\sec^{2}{x}$ but that doesn't help.
14
votes
2answers
799 views

Evaluate $\int\frac{1+x+\sqrt{1+x^2}}{\sqrt{x+1}+\sqrt{x}}\,dx$

Question: Solve the integral $$ \int\frac{1+x+\sqrt{1+x^2}}{\sqrt{x+1}+\sqrt{x}}\,dx $$ My solution: Multiply both the numerator and denominator by $\sqrt{x+1}-\sqrt{x}$. This changes the ...
14
votes
2answers
685 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot \log\left\{\frac{...
14
votes
1answer
529 views

Integral $\int{\frac{1}{\sqrt{2x^2+x+1}}}dx$

I am trying to solve this integral but I can not figure what I do wrong. $$I=\int{\frac{1}{\sqrt{(2x^2+x+1)}}}dx$$ Here's how I go about it: I think that maybe it can be solved following the $$\...