Questions tagged [indefinite-integrals]

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

-7
votes
0answers
68 views

Can someone help me to evaluate this triple integral that even Wolfram Alpha can't [on hold]

It's part of a classified project -- of a theory of everything in $n$-dimensions geometry. I spent 2 days trying to find calculators that shows the solution step-by-step, but I can't find anything... ...
0
votes
1answer
28 views

Why is this not a u-sub?

$\int e^*x dx$ Why is this not a u-sub? Where I let $u=x$ so $du=dx$ $\int e^udu = e^u +c$ I have notes where we did it as ab IBP problem instead.
1
vote
3answers
61 views

How to solve the indefinite integral?

The integral :- $$\int x^m \ln(a+x) \,dx.$$ (Also what is $m$ is not an integer, just an arbitrary real number?) I have found the integral in the book gradshteyn and ryzhik of which this is a ...
1
vote
1answer
64 views
-4
votes
2answers
44 views

Find $\int\sin^3(2x+1)dx$. [closed]

Find $\int\sin^3(2x+1)dx$. Having three different results which one is right right? 1.$y= -\frac{\cos (2x+1)}{2} +\frac{1}{24\cos(6x^3)} -\frac{1}{4\cos(2x+1)}+c$ $y=-\cos (2x+1) + \cos^3(2x+1) +c.$...
0
votes
1answer
43 views

Can u-substitution be used to solve integral where 'u' is NOT the inside function of a composite function?

I apologise in advance if this does not meet post guidelines. I am having difficulty with U-Substitution. I cannot seem to find an answer anywhere. Okay, so (if I'm not mistaken) u-substitution can ...
3
votes
0answers
66 views

Why this indefinite integral cannot be solved ?$\int\ln(x)\sqrt{x^{2 }+ ax + b}\, dx$ [closed]

I am trying to find a solution to this integral. However I have found nothing that helps, even wolfram alpha can not solve this integral. What would be the secret about it, since it looks so simple ? ...
1
vote
1answer
75 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}...
5
votes
4answers
204 views

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

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 ...
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 ...
1
vote
4answers
62 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!
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 \...
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
1answer
62 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 $\...
1
vote
2answers
60 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
2answers
89 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?
0
votes
3answers
78 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
55 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
57 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
114 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,+\...
-1
votes
2answers
55 views

Square indefinite integral calculation [closed]

How can we compute the integral? \begin{eqnarray} \int {\frac{\sqrt{x^2+x+1}}{x}} \end{eqnarray}
10
votes
0answers
102 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 ...
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
57 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
99 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( \...
1
vote
2answers
64 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
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
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 ...
3
votes
3answers
79 views

Evaluate $ \int \frac{x^4}{(2-x^2)^{3/2}}dx$

Evaluate the following integral: $ \int \frac{x^4}{(2-x^2)^{3/2}}dx$ I've tried to apply Chebyshev theorem on the integration of binomial differentials. We have $ m=4,a=2,b=-1,n=2,p=-3/2$. $\frac{...
2
votes
3answers
74 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\...
3
votes
0answers
84 views

Find f(x) using the fundamental theorem of calculus

Find $f(x)$ if $$\int_0^{f(x)}t^2dt=\pi \cos(\pi x)$$ So what I did is , using the fundamental theorem of calc. part 2 $$\int_0^{f(x)}t^2dt=\pi \cos(\pi x) \Leftrightarrow \frac{f(x)^3}{3}-0=\pi \cos(\...
0
votes
4answers
54 views

How to compute integral of $\frac{1}{5 - 4\sin(x) + 3\cos(x)}$

So, I need to compute a integral for $$ \frac{1}{5-4 \sin(x) + 3 \cos(x)}. $$ On integral calculator the following transformation is made: $$ -\frac{\sec^2\left(\frac{x}{2}\right)}{2\left(\tan\left(\...
1
vote
2answers
46 views

The integral of $5/\left(x^2+2\right)$

I have to calculate a integral for following equation: $\frac{5}{x^2+2}$. On the integral calculator they show that it must be solved by substitution and the substitution must be $u=\frac{x}{\sqrt{2}}$...
1
vote
3answers
99 views

Why doesn't the substitution $x=it$ work in the integral $\int\frac{x^2}{{(x^2+1)}^2}dx$?

For İntegral $$\int\frac{x^2}{{(x^2+1)}^2}dx$$ I used this substitution $x=it$ We have, $$\begin{align} \int\frac{x^2}{{(x^2+1)}^2}dx &=-i\int\frac{t^2}{{(t^2-1)}^2}dt \\ &=-\frac i4\left(...
1
vote
1answer
35 views

Computing anti-derivative of this function

Consider the function on the real numbers $f(x):=\arctan(\frac x{\sqrt{1+x^4}})$. One can easily find that $f’(x)=\frac{1-x^4}{(1+x^2+x^4)\sqrt{1+x^4}}$. I was wondering how one could compute the ...
1
vote
1answer
39 views

Integral of cosine to integer powers

I found the following on Wolfram Math World: I understand the first step, the integration by parts. For the first case, m is even, I am pretty sure it involves a binomial expansion of $(\frac{1}{2}(1-...
3
votes
2answers
68 views

An indefinite Integral Problem with algebric numerator and trigonometric denominator

$$\int \frac{x^2+(n(n-1))}{(x\sin x +n\cos x)^2 } dx$$ I know this is an homework problem, but I really couldn't think of any way to solve it. Like DI Method (No go) , What kind of substitution as ...
1
vote
0answers
69 views

Proof-Verification:$\int x[3+\ln(1+x^2)]\arctan x{\rm d}x$.

$$\begin{aligned} &\int x[3+\ln(1+x^2)]\arctan x{\rm d}x\\ =&\int 3x\arctan x{\rm d}x+\int x\ln(1+x^2)\arctan x{\rm d}x\\ =&\int \arctan x{\rm d}\left(\frac{3x^2}{2}\right)+\int \ln(1+x^2)\...
3
votes
1answer
155 views

How to evaluate $\int\sqrt{1-\tan x} \,dx$ without using up paper?

$$\int\sqrt{1-\tan x} \, dx$$ is a very interesting integral. I attempted to evaluate it with the substitution $u^2=1-\tan x$ and then obtaining partial fractions. However, the coefficients are ...
5
votes
1answer
52 views

Check my work: General solution of a PDE

I have been asked to find the "Most general solution" for $u(x,y)$ of the PDE $$\frac{\partial u}{\partial x} = \frac{x}{\sqrt{x^2+y^2}} + 3y\cos(3xy) + 3x^2y^2$$ I know you must take the integral ...
0
votes
1answer
48 views

integration with trigonometric substitution, is my result correct?

$$\int\frac{\ln x}{x\sqrt{1-4\ln x-\ln^{2}x}}\ dx\left | u=\ln x,\ du=\frac{1}{x},\ x du=dx \right | \\ \int \frac{u}{x\sqrt{-u^{2}-4u+1}}\ xdu\ = \int \frac{u}{\sqrt{-u^{2}-4u+1}}\ du \\ \left | -u^{...
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
31 views

Integrating $\int \left(\sqrt[6]{\frac{x}{x-2}} - \sqrt[4]{\frac{x}{x-2}}\right)\frac{\mathrm dx}{x^2-2x}$ with partial fractions

Recently I've been studying on partial fractions and integration using partial fraction decomposition. I've not had any problems solving those types of integrals until I came across this integral: ...
2
votes
1answer
52 views

Find $\int\frac{dx}{p\left(e^x\right)}$ where $p$ is a polynomial

Currently I'm facing a lot of integrals of the form $$I(p)=\int\frac{dx}{p\left(e^x\right)}$$ where $p:\mathbb{R}\to\mathbb{R}$ is a polynomial. For example, $$\begin{split}I(x+1)&=\int\frac{dx}{e^...
0
votes
1answer
50 views

Does the integral $\int \frac{1}{(ax+b)(cx+d)} dx$ converge?

Let $a, b, c, d \in \mathbb{R}$. I was wondering does the integreal $\int_A^{\infty} \frac{1}{(ax+b)(cx+d)} dx$ converge? where the integrand is well defined for $x\geq A$? I think it should ...
2
votes
1answer
74 views

Evaluate integral of $\ln(\sqrt{x^2+1}+x)$ [duplicate]

Evaluate: $\displaystyle\int\limits^{\cssId{upper-bound-mathjax}{\class{placeholder}{}}}_{\cssId{lower-bound-mathjax}{\class{placeholder}{}}} \ln\left(x+\sqrt{1+x^2}\right)\,\cssId{int-var-mathjax}{\...
0
votes
3answers
65 views

Integrate by parts - $\int\sqrt{a^2-x^2}dx$

Differentiate $\arcsin\left(\dfrac{x}{a}\right)$ with respect to x. Integrate by parts: $\int\sqrt{a^2-x^2}dx$ The answer to part one of the question is $\dfrac{1}{\sqrt{a^2-{x^2}}}$
1
vote
2answers
41 views

indefinite integral of 1/x where x is dimensional?

As we know, the integral of $\frac{1}{x}$ is $ln(x)+c$. Because $x$ and $dx$ have the same dimension, $\int\frac{dx}{x}$ is dimensionless. But my problem is: $x$ is dimensional. I've been trained that ...