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Questions tagged [indefinite-integrals]

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

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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 ...
148
votes
19answers
24k 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 ...
94
votes
11answers
9k 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 ...
50
votes
6answers
48k 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. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$
18
votes
2answers
24k 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 ...
6
votes
20answers
796 views

Integration of $\int\frac{1}{x^{4}+1}\mathrm dx$ [closed]

I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}\mathrm dx$. Do I have to use trigonometric substitution?
82
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 ...
12
votes
3answers
94k views

Integral of $\sqrt{1-x^2}$ using integration by parts

I was asked to solve this indefinite integral using Integration by parts. $$\int \sqrt{1-x^2} dx$$ I know how to solve if use the substitution $x=\sin(t)$ but I'm looking for the Integration by ...
18
votes
4answers
525 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?
10
votes
3answers
3k views

What is the integration of $\int 1/(x^{2n} +1)dx$?

I am a student who is preparing for IIT exam. I was just practicing calculus and encountered this problem. I tried different substitutions but none of them seemed to work. So what is the integration ...
214
votes
15answers
29k 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? ...
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 ...
8
votes
3answers
642 views

Solving this integral $\int\frac{1}{1+x^n} dx$?

Well, this might be a really simple one. But still... What will be the soln. to --- \begin{aligned} \int\frac{1}{1+x^n} dx \end{aligned} Is substituting \begin{aligned} 1+x^n \end{aligned} by tan z ...
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 ...
33
votes
3answers
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&...
11
votes
10answers
514 views

How do I evaluate $\int \frac{\mathrm{d}x}{e^x + 1} $?

How do I solve evaluate $$\int \frac{\mathrm{d}x}{e^x + 1}\ ?$$ I know that I have to use $u$ substitution but I can't seem to find something to substitute with.
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 ...
12
votes
7answers
1k views

Indefinite Integral with “sin” and “cos”: $\int\frac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $

Indefinite Integral with sin/cos I can't find a good way to integrate: $$\int\dfrac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $$
5
votes
2answers
635 views

Indefinite integral: $\int \frac{\mathrm dx}{\sqrt x(1+\sqrt[3]x)}$

I need : $$ \int \frac{\mathrm dx}{\sqrt{x}(1+\sqrt[3]{x})}$$ Can you give me a hint, which way would suit best? I tried substitution, but don't have an idea what to substitute.. :/
19
votes
5answers
102k 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 ...
3
votes
4answers
9k views

Calculate $\int\left( \sqrt{\tan x}+\sqrt{\cot x}\right)dx$ [duplicate]

How to calculate following integration? $$\int\left( \sqrt{\tan x}+\sqrt{\cot x}\right)dx$$
8
votes
1answer
289 views

How to evaluate these indefinite integrals with $\sqrt{1+x^4}$?

These integrals are supposed to have an elementary closed form, but Mathematica only returns something in terms of elliptic integrals. I got them from the book Treatise on Integral Calculus by Edwards....
6
votes
1answer
529 views

How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int e^{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-\...
48
votes
6answers
10k views

What is the antiderivative of $e^{-x^2}$

I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$ So, I ...
28
votes
2answers
9k 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 ...
13
votes
4answers
7k views

Evaluate $\int \cos(\cos x)~dx$

Evaluate $\int \cos(\cos x)~dx$ I tried to use chain rule but failed. Can anyone help me please?
12
votes
3answers
5k views

Integral $\int\!\sqrt{\cot x}\,dx $

Find the integral $$\int\!\sqrt{\cot x}\,dx $$ How can one solve this using substitution? Can this be solved by complex methods?
13
votes
3answers
20k views

How to integrate $\int \frac{1}{\sin^4x + \cos^4 x} \,dx$?

How to integrate $$\int \frac{1}{\sin^4x + \cos^4 x} \,dx$$ I tried the following approach: $$\int \frac{1}{\sin^4x + \cos^4 x} \,dx = \int \frac{1}{\sin^4x + (1-\sin^2x)^2} \,dx = \int \frac{1}{\...
7
votes
3answers
9k views

How to integrate $\int e^{-t^{2}} \space \, \mathrm dt $ using introductory calculus methods

Earlier today I stumbled across this when I was doing some practice questions for a physics course: $$\int e^{-t^2} \space \, \mathrm dt $$ To expand, the limits of integration were something like $...
2
votes
1answer
63 views

Integration of elementary rational fractions

$$\int{\frac{Mx+N}{x^2+px+q}}dx$$ When all the coefficients are given, it can be easily solved using substitution but I have no idea how to find a solution in a standard form. What are the first ...
1
vote
1answer
166 views

Generalized Owen's T function

As Wikipedia teaches us https://en.wikipedia.org/wiki/Owen%27s_T_function the Owen's T function $T(h,a)$ defines a probability of a bivariate event $X>h$ and $0<Y<a X$ where $X,Y$ are ...
0
votes
6answers
12k views

Evaluating $\int \sqrt{1 + t^2} dt$?

How do we solve (i.e. get the closed form of) $\int \sqrt{1 + t^2} dt$ ? The Wolfram page shows the closed form of it but not the steps in solving it. I think I need some algebraic trick.. I ...
12
votes
4answers
507 views

Help with $\int\frac{1}{1+x^8}dx$

I'm not sure how to proceed. I tried factoring like you do to evaluate $\int\frac{1}{1+x^4}dx$, and since it came out nasty I checked out WolframAlpha to see if I was on the right track. In fact, ...
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
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 ...
10
votes
2answers
3k views

Need help solving - $ \int (\sin 101x) \cdot\sin^{99}x\,dx $

I have a complicated integral to solve. I tried to split ($101 x$) and proceed but I am getting a pretty nasty answer while evaluating using parts. are there any simpler methods to evaluate this ...
8
votes
4answers
14k views

What is the integral of $e^{\cos x}$

Question: Find out $\displaystyle{\int e^{\cos x}~dx}$. My Attempt: Let $\cos x = y$. Hence $-\sin x\ dx = dy$ or $$dx = \displaystyle{\frac{-dy}{\sin x}=\frac{-dy}{\sqrt{1-\cos^2x}}=\frac{-dy}{\...
7
votes
4answers
40k views

Integration of sqrt Sin x dx

$$\int \sqrt{\sin x}\ \text dx$$ I asked my teachers and they said that this integration is pretty next level and will be taught in college. Can anyone help?
3
votes
1answer
145 views

Another way to evaluate $\int\frac{\cos5x+\cos4x}{1-2\cos3x}{dx}$?

What I've done is this:$$\int\dfrac{\cos5x+\cos4x}{1-2\cos3x}{dx}$$ $$\int \dfrac{\sin 3x}{\sin 3x}\left[\dfrac{\cos5x+\cos4x}{1-2\cos3x}\right]{dx}$$ $$\dfrac {1}{2}\int\dfrac{\sin 8x -\sin 2x +\sin ...
5
votes
5answers
1k views

Finding $\int e^{2x} \sin{4x} \, dx$

Finding $$\int e^{2x} \sin 4x \, dx$$ I think I should be doing integration by parts... If I let $u=e^{2x} \Rightarrow du = 2e^{2x}$, $dv = \sin{4x} \Rightarrow v = -\frac{1}{4} \cos{4x}$ $\int{ e^{...
4
votes
2answers
4k views

How do you integrate $e^{x^2}$?

I know that $\int{\frac{1}{x}}dx$ is simply $\ln{(x)}+c$ (-which is clearly unrelated to the problem but I just thought I would share anyway) but I am not sure how to approach $e^{x{^2}}$. Perhaps a ...
5
votes
0answers
374 views

Integral of $\frac{1}{x^4+1}$ [duplicate]

Just doing this for revision, seems much harder than it should be, should I use $x=\tan u$ ? Any help appreciated.
11
votes
3answers
865 views

calculation of $\int\frac{1}{\sin^3 x+\cos^3 x}dx$ and $\int\frac{1}{\sin^5 x+\cos^5x}dx$

Solve the following indefinite integrals: $$ \begin{align} &(1)\;\;\int\frac{1}{\sin^3 x+\cos^3 x}dx\\ &(2)\;\;\int\frac{1}{\sin^5 x+\cos^5 x}dx \end{align} $$ My Attempt for $(1)$: $$ \...
7
votes
3answers
679 views

So close yet so far Finding $\int \frac {\sec x \tan x}{3x+5} dx$

Cruising the old questions I came across juantheron asking for $\int \frac {\sec x\tan x}{3x+5}\,dx$ He tried using $(3x+5)^{-1}$ for $U$ and $\sec x \tan x$ for $dv$while integrating by parts. below ...
11
votes
4answers
531 views

How to evaluate $\int\frac{1+x^4}{(1-x^4)^{3/2}}dx$?

How do I start with evaluating this- $$\int\frac{1+x^4}{(1-x^4)^{3/2}}dx$$ What should be my first attempt at this kind of a problem where- The denominator and numerator are of the same degree ...
3
votes
1answer
277 views

Today a student asked me $\int \ln (\sin x) \, dx.$

Calculate the integral $$\int \ln (\sin x) \, dx.$$
3
votes
2answers
927 views

Given an exact differential; $df=yz\,dx+xz\,dy+(xy+a)\,dz$: Why must we integrate each term independently to find the parent function $f\,$?

In other words; Why can't I integrate the whole equation in one go like this? $$\begin{align}f=\int df&=\int yz\,dx +\int xz\,dy+\int xy\,dz+\int a\,dz\\&=xyz+xyz+xyz+az+C\\&=3xyz + az +C\...
8
votes
6answers
236 views

Two apparently different antiderivatives of $\frac{1}{2 x}$ [duplicate]

What is right way to calculate this integral and why? $$ \int\frac{1}{2x}\text dx $$ I thought, that this substitution is right: $$ t = 2x $$ $$ \text dt = 2\text dx $$ $$ \frac{\text dt}{2} = \...
6
votes
2answers
189 views

Evaluate $\int\frac{1}{1+x^6} \,dx$

I came across following problem Evaluate $$\int\frac{1}{1+x^6} \,dx$$ When I asked my teacher for hint he said first evaluate $$\int\frac{1}{1+x^4} \,dx$$ I've tried to factorize $1+x^6$ as $$...