Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [indefinite-integrals]

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

1
vote
1answer
26 views

SImplification of $I=\int_{x=0}^{\infty}x^{n-1}(\alpha-x)^{m}e^{-\mu x}dx.$

Let $n$ and $m$ be positive integers, $\mu$ be real positive and $\alpha$ positive real number. I would like to compute the following integral if there is close formula $$ I=\int_{x=0}^{\infty}x^{n-...
0
votes
0answers
19 views

Relating two integrals

Is there a way to relate this integral $$ \int_{-\infty}^{\infty}\prod_{i=1}^{n}\left(\frac{1}{1+c_{i}e^{-b_{i}z}}\right)^{\alpha_{i}}\left(1-\frac{1}{1+c_{i}e^{-b_{i}z}}\right)^{\beta_{i}}\mathrm{d}...
2
votes
2answers
88 views

Evaluating $\displaystyle\int \frac{\cos x}{\sin^3x+\sin x}dx$

Given the function $$g(x)=\frac{\cos x}{\sin^3x+\sin x}$$, by letting $u=\sin x$, show that $$\int g(x) dx=\int\left(\frac{A}{u}+\frac{Bu+C}{u^2+1}\right)du$$ where $A,B$ and $C$ are constants. Hence, ...
2
votes
4answers
129 views

Unable to solve $ \int \frac{x + \sqrt{2}}{x^2 + \sqrt{2} x + 1} dx $?

This comes from a bigger problem :- $$ \text{Evaluate } \int\frac{dx}{1+x^4} $$ After making $ \int \frac {dx}{1+x^4} = \frac{dx}{(1+x^2)^2 - (\sqrt{2}x)^2} $ and then applying partial fraction ...
1
vote
1answer
74 views

The integral of x is not the same as $x e^{- k x}$ if k goes to $0$ .

I have this problem, I don't know why if I consider: $k \geq 0$ $ I(k)= \int x e^{-k x}= - \dfrac{e^{-kx}(kx+1)}{k^2}$, with integration constant $c=0$. If I put ${lim}_{k \to 0} I(k) {\to -\infty}...
0
votes
0answers
26 views

Primitive of a function for all but countable many points

Let $f$ be a real-valued function with $D(f) = D = <a,b>$. In our calculus course we introduce such definitions: Definition 1. $F$ is an exact primitive of $f$ iff $D(F) = D$, $F$ is ...
-2
votes
0answers
32 views
3
votes
2answers
310 views

Integration of $ \int \frac{dx}{x^2\sqrt{x^2 + 9}} $ using trigonometric substitution

I have been stuck trying to figure out an integration problem involving trigonometric substitution. $$ \int \frac{dx}{x^2\sqrt{x^2 + 9}} $$ So I substituted $$ x = 3\tan\theta $$ $$ dx = 3\sec^2\...
1
vote
4answers
60 views

Find the derivative of $F(x)=\int_{\pi}^{\ln x} \cos e^t dt$

Am I supposed to change the limits of integration? $$F(x)=\int_{\pi}^{\ln x} \cos e^t dt = \int_{e^\pi}^x \frac{\cos u}{u} du $$ Help!
0
votes
1answer
15 views

Going backwards from derivative value, to retrieve point and order.

Using Sage let's say I do the following. Take the $4^{th}$ order derivative of $\sin(x)$ about the point $2$: ...
1
vote
3answers
149 views

Integrate $\int \dfrac{e^x}{1+\cos x}dx$

Integrate $\displaystyle\int \dfrac{e^x}{1+\cos x}dx$ My Effort; I couldn't nothing.
6
votes
2answers
52 views

How is Wolfram Alpha and the reduction formula arriving at a different result for the integral of $\int \sec^4 x\,dx$ than naive $u$-substitution?

I calculated the following on paper for the value of $\int \sec^4 x\,dx$. $$\int \sec^4 x\,dx=\int \sec^2 x \sec^2 x\,dx=\int (\tan^2 x + 1)(\sec^2 x)\,dx.$$ Let $u = \tan x$, $du = \sec^2 x\,dx$ so \...
10
votes
0answers
98 views

Symbolic approximation through integration by parts

This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt $ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get ...
2
votes
1answer
171 views

Integrate $\sec^n(x)\ dx$ for odd n, without using a recursive formula.

$$\int \sec^{2n+1}(x)\ \text{d}x $$ This is not a homework. I would like someone to suggest a solution for this, without using any recursion formula.
1
vote
1answer
61 views

Integral of $a^x\cos^axdx$

I was given the following integral to solve: $$\displaystyle \int 3^x\cos^3x dx$$ Writing $\cos^3x = \dfrac{\cos 3x + 3\cos x}{4}$ and $3^x = e^{x \ln 3}$, and using the standard result $\...
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 $...
12
votes
6answers
30k views

How to integrate a three products

I tried to integrate $x e^x \sin x$, using integration by parts, and setting $\frac{\, \mathrm dv}{\, \mathrm dx} = e^x \sin x$. Even though I got really close, I kept getting it wrong. Can someone ...
13
votes
1answer
333 views

Finding indefinite integral $\int{ \mathrm dx\over \sqrt{\sin^3 x+\sin (x+\alpha)}}$

Could anyone help me how to solve this indefinite integral? $$\int{\mathrm dx\over \sqrt{\sin^3 x+\sin (x+\alpha)}}$$
3
votes
1answer
602 views

evaluate $\int\ln x\tan x\, \mathrm dx$

How to evaluate $$\int\ln x\tan x\, \mathrm dx \:?$$ I've tried to do integration by parts but after calculations it cancel out the main question.
6
votes
2answers
318 views

evaluate $\int \frac{\tan x}{x^2+1}\:dx$

$$\int \frac{\tan x}{x^2+1}\, \mathrm dx$$ I used By-parts method setting $u=\tan x$ and $\, \mathrm dv=\frac{1}{x^2+1}\, \mathrm dx$, but then I got an integral that's more complicated I also ...
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 ...
1
vote
2answers
57 views

How to integrate $\sqrt{\arctan(x)}$ [closed]

How to do $$\int\sqrt{\arctan(x)}\, \mathrm dx \:??? $$ Is there any other special function defined like this?
0
votes
3answers
77 views

Verifying an indefinite integral solution

I'm working on indefinite integrals right now. I'm given the problem $$\int(\frac{8}{x}-\frac{5}{x^2}+\frac{6}{x^3}) dx$$ My worksheet gives the answer $8\ln(x)+\frac{5}{x}-\frac{3}{x^2}+c$ I'm ...
0
votes
2answers
88 views

Indefinite integration of $\int \frac{1}{1+\sqrt{x^2+2x+2}}dx$

Integrate $$\int \frac{1}{1+\sqrt{x^2+2x+2}}dx$$ I have tried by using Euler substitution, but that gave me a wrong answer. So can somebody help?
4
votes
4answers
226 views

Is $\int\frac{\mathrm d \, x}{x^2-1}= \operatorname{arctanh} ~x + C = \operatorname{arccoth} ~x + K$ correct?

My professor gave us an integral but I think it's incorrect. Is this correct? $$ \DeclareMathOperator\arctanh{arctanh} \DeclareMathOperator\arccoth{arccoth} \int\frac{\mathrm d \, x}{x^2-1}= \arctanh ...
0
votes
2answers
53 views

Evaluate $\int \frac{1}{1+3\sin^2 x} dx$ (Making antiderivative continuous.)

Evaluate $\int \frac{1}{1+3\sin^2 x} dx$ I know that this has an antiderivative on $\mathbb{R}$ I can use the trig. substitution $t = \tan x$ on $(-\frac{\pi}{2}+k\pi, \frac{\pi}{2}+k\pi)$ $x = \...
3
votes
1answer
54 views

Mistake in evaluating the secant integral?

I was trying to solve the secant integral $$\int \dfrac{1}{\cos x} dx $$ by using the substitution $t := \tan(\dfrac{x}{2})$. Using this, I found: $$dx =\dfrac{2\cdot dt}{t^2 + 1} $$ and $$\cos(x)...
2
votes
1answer
113 views

What's $ \int \frac{1}{2+\cos 2x}dx$ on earth?

Let's consider a problem, which is to find the indefinite integral $$I(x):=\displaystyle \int \frac{1}{2+\cos 2x}dx.$$ Since the integrand $f(x):=\dfrac{1}{2+\cos 2x}$ is continuous over $(-\infty,+\...
2
votes
1answer
48 views

Elliptic Integral-ish?

I'm trying to solve this integral $\int (1-\cos(\theta))^{2}\sqrt{1-k^{2}cos(\theta)^{2}} \mathrm{d}\theta$ I think it's some kind of elliptic integral but i can't integrate.
-1
votes
2answers
53 views

Square indefinite integral calculation [closed]

How can we compute the integral? \begin{eqnarray} \int {\frac{\sqrt{x^2+x+1}}{x}} \end{eqnarray}
3
votes
1answer
74 views

Evaluate $\int \frac{2\pi y}{2y^3-1}\,dy$

Evaluate: $$\int \dfrac{2\pi y}{2y^3-1}\,dy$$ I've been struggling with this for a while. If it had just been $y^3$ instead of $2y^3$ in the Denominator, Partial Fraction Decomposition, although ...
1
vote
1answer
64 views

Evaluating this integral $\int \frac{3x^2-1}{2x\sqrt{x}}\arctan(x){\rm d}x$, how to start?

I would like to evaluate $$\int \dfrac{3x^2-1}{2x\sqrt{x}}\arctan(x){\rm d}x$$ I'm not even sure how to start, any suggestion will be appreciated.
2
votes
3answers
86 views

Evaluate $\int\frac{da}{a\sqrt{a+1}}$

$$\int\dfrac{da}{a\sqrt{a+1}}$$ I don't know how to solve this integral. The fact that $\dfrac1a$ is the derivative of $\ln(a)$ and $\dfrac{1}{\sqrt{a+1}}$ is the derivative of $\cos^{-1}a$ suggested ...
7
votes
3answers
199 views

Trignometric integral : $\int \frac{dx}{\sin x + \sec x}$

The integral I am trying to compute is : $$\int \dfrac{dx}{\sin x + \sec x}$$ I have tried manipulating trignometric functions and it took me nowhere. Then finally I tried putting $\tan\dfrac{x}{2} =...
3
votes
3answers
146 views

Find $\int \frac{\sqrt{1-x^2}}{1+x^2}\hspace{1mm}dx$

Find $$\int \dfrac{\sqrt{1-x^2}}{1+x^2}\hspace{1mm}dx$$ Any hints! I will do the work, just give me a clue
2
votes
2answers
112 views

Fnd $ \int\frac{e^{-2x-x^2}}{( x+1)^2}\,dx$

Find $\displaystyle\int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$. If I do Integration by parts, I end up with $\displaystyle\int e^{-2x-x^2}\hspace{1mm}dx$ Which I believe cannot be ...
7
votes
3answers
418 views

Evaluate the integral: $ \int x \tan^{-1}\ x \,\mathrm{d}x$

Evaluate the integral: $$\int x\tan^{-1}x\,\mathrm{d}x$$ What I have so far: $$u = \tan^{-1}x$$ $$\mathrm{d}u = \frac{1}{1+x^2}\,\mathrm{d}x$$ $$\mathrm{d}v = x\,\mathrm{d}x$$ $$v = \frac{x^2}2$$ $...
2
votes
0answers
149 views

Integral of $\int \sin^{-1} \left (e^{\sqrt x}\right )\, \mathrm{d}x$

How do I evaluate the following; $$\displaystyle \int \sin^{-1} \left (e^{\sqrt x}\right ) \mathrm{d}x$$ $$\displaystyle \int \sin^{-1} \left (e^{-\sqrt x} \right )\mathrm{d}x$$ Is there a closed ...
2
votes
2answers
97 views

Find the integral of $\int \ln\left(\sqrt{x-b}+\sqrt{x-a}\right)\,dx$

Evaluate $\displaystyle\int \ln\left(\sqrt{x-b}+\sqrt{x-a}\right)\,dx$. I am tryed to integrate it by parts by taking $du = 1$ and $v=\ln\left(\sqrt{x-b}+\sqrt{x-a}\right)$ Therefore, $vu - \...
0
votes
2answers
89 views

Evaluate $ \int \frac{a^2\cos^2x+b^2\sin^2x}{a^4\cos^2x+b^4\sin^2x}\,dx$

Evaluate $$ \int \frac{a^2\cos^2x+b^2\sin^2x}{a^4\cos^2x+b^4\sin^2x}\,dx$$ I have tried Weierstrass substitution and tried to split into two integrations, but it gets really messy. Is there a better ...
0
votes
0answers
55 views

Solving $\int dx x^{-\beta} \zeta(\beta, 1 + 1/x)$

The integral is indefinite, $\zeta(\cdot, \cdot)$ is the Hurwitz function and $\beta > 1$ is a constant. When $\beta = 2$, the integral results in the digamma function $-\psi \left(1 + \frac{1}{x}...
8
votes
2answers
98 views

Difference in my and wolfram's integration.

Calculate $$\int \frac{\sin ^3(x)+1}{\cos (x)+1} \, dx$$ Let $$u = \tan(x/2)$$ $\int \frac{\sin ^3(x)+1}{\cos (x)+1} \, dx = \int \frac{2\left(\frac{8u^3}{(u^2+1)^3}+1 \right)}{(u^2+1)\left( \...
2
votes
3answers
150 views

The Integral $\int \frac {dx}{(x^2-2ax+b)^n}$

Recently I came across this general integral, $$\int \frac {dx}{(x^2-2ax+b)^n}$$ Putting $x^2-2ax+b=0$ we have, $$x = a±\sqrt {a^2-b} = a±\sqrt {∆}$$ Hence the integrand can be written as, $$ \frac {1}...
1
vote
2answers
63 views

Solve integral $\int_0^\infty \sqrt{x}e^{-\sqrt[3]{x}} dx$

$$\int_0^\infty \sqrt{x}e^{-\sqrt[3]{x}} dx$$ I have tried solving it by substituting $ x=u^2 $ but couldn't solve it.
1
vote
3answers
62 views

Definite Integral of $\int_0^1\frac{dx}{\sqrt {x(1-x)}}$

We have to calculate value of the following integral : $$\int_0^1\cfrac{dx}{\sqrt {x(1-x)}} \qquad \qquad (2)$$ What i've done for (2) : \begin{align} & = \int_0^1\cfrac{dx}{\sqrt {x(1-x)}} \\ &...
1
vote
3answers
58 views

Integrate $\int \frac{dx}{a\sin x+b\cos x}$

As far as I know, we could use the stereographic change of variables where $\tan(\frac{x}{2})=t$, $\sin x=\frac{2t}{1+t^2}$ and $\cos x= \frac{1-t^2}{1+t^2}$, then replace $dx$ also $\sin x$ and $\...
1
vote
1answer
41 views

Why doesn't a constant appear when solving $\int{e^x \sin(x)dx}$?

$\int e^x\sin(x)dx$ $= e^x\sin(x) - \int e^x\cos(x)dx$ $\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\...
1
vote
2answers
75 views

Suppose that the average value on all intervals $[a,b]$ is equal to $f((a+b)/2)$. Prove that $f''(x) = 0$ for all $x \in \mathbb{R}$

I understand that $f(x)$ must be linear with a first derivative equal to a constant. I'm just not sure how I can use the mean value property of integrals to show something about $f''(x)$. The hint on ...
2
votes
3answers
73 views

Evaluate $\int \frac{dx}{\left(\frac{1}{x}-\frac{1}{a}\right)}$

Evaluate the following integral: $$\displaystyle \int\dfrac{dx}{\left(\dfrac{1}{x}-\dfrac{1}{a}\right)}$$ Where $a$ is an arbitrary constant. How do I solve this? I tried the substitution $$x=a\...
2
votes
2answers
79 views

Evaluate $\int \frac{dx}{\sqrt{\frac{1}{x}-\frac{1}{a}}}$

Evaluate the following integral: $$\displaystyle \int\dfrac{dx}{\sqrt{\dfrac{1}{x}-\dfrac{1}{a}}}$$ Where $a$ is an arbitrary constant. How do I solve this? EDIT: I would appreciate it if you ...