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Questions tagged [indefinite-integrals]

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

-4
votes
1answer
29 views

Problems related to Indefinite Integrals.Higher Secondary School Mathematics. [on hold]

What are the methods i can use to solve this question? $$\int \frac{ \sqrt{x^2 +1} - \sqrt{x^2 -1} }{ \sqrt{x^4 -1} }$$ How to find this?
3
votes
1answer
77 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
52 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
44 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
91 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
24 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
36 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
42 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
1answer
61 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
145 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
46 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
88 views

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

Evaluate $\displaystyle\int \ln\left(\sqrt{x-b}+\sqrt{x-a}\right)$ 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
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2answers
28 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: ...
-1
votes
0answers
35 views

Proof-Verification: Find $\int e^{-\frac{\pi}{2}t^4}t^4 {\rm d}t$.

Solution Integrating by parts,one can obtain \begin{align*} \int e^{-\frac{\pi}{2}t^4}t^4 {\rm d}t&=-\frac{1}{2\pi}\int t {\rm d}\left(e^{-\frac{\pi}{2}t^4}\right) =-\frac{1}{2\pi}te^{-\frac{\pi}{...
2
votes
1answer
51 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
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1answer
49 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
71 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
63 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
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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 ...
0
votes
0answers
31 views

Calculate integral involving upper incomplete Gamma function

I need to calculate the following integral: $\int_{z/x}^{0}e^{-\frac{y}{b_n}}y^l\Gamma(c,ky)dy$, with $z<0,z\in R,x>0,x\in R,b_n>0,b_n \in R,l \geq0,l\in Z,k<0,k\in R,c\geq1,c\in Z$ I ...
0
votes
3answers
75 views

How do I find $\int e^{x \sin x +\cos x}(\frac{x^4 \cos^3 x - x \sin x + \cos x}{x^2 \cos ^2 x})\,dx$?

How do I find $$\int e^{x \sin x +\cos x}\left(\frac{x^4 \cos^3 x - x \sin x + \cos x}{x^2 \cos ^2 x}\right) dx\quad?$$ I tried to put $e^{x \sin x +\cos x}$ as some t and write the expression in ...
2
votes
0answers
49 views

Does $\int \frac{e^x(x-1)}{1+xe^x}dx$ have any closed form?

It's simple that $$\int \frac{e^x(x+1)}{1+xe^x}dx=\ln(1+xe^x)+C,$$ But what if $$\int \frac{e^x(x-1)}{1+xe^x}dx?$$
4
votes
2answers
54 views

Indefinite integral through factorization

So I have $$\int \frac{1}{(x^2+1)^2}dx$$ And the professor does some magic I'm confused. what's with the derivative? I solved the integral via substitution but I'm curious how this works, so I can ...
1
vote
0answers
57 views

Evaluation of the integral $ \int \frac{x^{\frac{1}{3}}}{1+x^3 } dx $ [closed]

I'm looking to solve this integral right here: $ \int \frac{x^{\frac{1}{3}}}{1+x^3 } dx $ I would like to know what approaches I could take to solve this using complex analysis.
0
votes
2answers
63 views

Integral $ \int_\sigma^\infty r^2 {e^{-A/r^6}} dr $ [closed]

Below integral can be calculated by using taylor expansion for the $ e^{-A/r^6} $ term. I want to know how to solve this integral analytically? $$ \int_\sigma^\infty r^2 {e^{-A/r^6}} dr $$ Hint: I ...
2
votes
2answers
105 views

Computing the integral $x^2\textrm{sech}^2(x)$

I'm trying to compute the integral $$\int_{0}^{\infty}dx \, x^{2}\operatorname{sech}^{2}(x)=\frac{\pi^{2}}{12}.$$ Manually, one obtains, quite naively, $$\int dx \, x^{2}\operatorname{sech}^{2}(x)=\...
3
votes
1answer
77 views

Evaluate $ \int \left(\frac{1}{\sqrt[3]x +\sqrt[4]x} + \frac{\ln(1+\sqrt[6]x)}{\sqrt[3]x +\sqrt x}\right)\,dx$

Evaluate the following integral: $$\int \left(\frac{1}{\sqrt[3]x +\sqrt[4]x} + \frac{\ln(1+\sqrt[6]x)}{\sqrt[3]x +\sqrt x}\right)\,dx$$ My Attempt: Let $$I=\int \Big(\frac{1}{\sqrt[3]x +\sqrt[4]...
7
votes
3answers
116 views

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

Integrate $$\int \frac {\sin (2x)}{(\sin x+\cos x)^2} \,dx$$ My Attempt: $$=\int \frac {\sin (2x)}{(\sin x + \cos x)^2} \,dx$$ $$=\int \frac {2\sin x \cos x}{(\sin x+ \cos x)^2} \,dx$$ Dividing the ...
0
votes
4answers
65 views

Integrate $\int \frac {dx}{a+b\sin x}$ where $a^2<b^2$

Integrate: $$\int \dfrac {dx}{a+b\sin x} \quad \,, \,a^2<b^2$$ My Attempt: $$=\int \dfrac {dx}{a + b\dfrac {2\tan (\dfrac {x}{2})}{1+\tan^2 (\dfrac {x}{2})}}$$ $$=\int \dfrac {(1+\tan^2 (x/2)) dx}...
-1
votes
3answers
69 views

What am I doing wrong in evaluating $\int {\frac {dx}{1+x-x^2}}$?

This is my solution :- $$\int \frac{dx}{1 + x - x^2} = \int \frac{dx}{-(x^2 - x - 1)} = -\int \frac{dx}{x^2 - x - 1} = -\int \frac{dx}{x^2 - 2.x.\frac{1}{2} + (\frac{1}{2})^2 - \frac{5}{4}} = -\int \...
1
vote
1answer
26 views

Compute $\int_{t-2T}^T t-\tau\cdot 1 \;d\tau = [t\tau - \frac{\tau^2}{2}]_{t-2T}^{T}$

Compute $\displaystyle\int\limits_{t-2T}^T t-\tau\cdot 1 \;d\tau$ \begin{align} \displaystyle&\int\limits_{t-2T}^T t-\tau\cdot 1 \;d\tau \\ &=\left[t\tau - \frac{\tau^2}{2}\right]_{t-2T}^{T}\\...
2
votes
1answer
32 views

u-substitution yields a different answer

When I compute the indefinite integral of ln(x + x^2), I get 2 answers from 2 different methods. First method: integration by parts => u-substitution Answer = xln(x + x^2) - 2 (x + 1) + ln l x + 1 l ...
1
vote
5answers
79 views

Finding interesting ways to solve $\int \frac{2\sinh(x)+1}{2\cosh(3x)} {\rm d}x$.

$$\int \frac{2\sinh(x)+1}{2\cosh(3x)} {\rm d}x \\ = \int \frac{e^x + 1 - e^{-x}}{e^{3x} + e^{-3x}} {\rm d}x \\ = \int \frac{e^{4x} + e^{3x} - e^{2x}}{e^{6x} + 1} {\rm d}x \\ = \int \frac{u^3 + u^2 -u}{...
1
vote
1answer
75 views

Evaluating $\int \frac{1}{\sqrt{5+4\cos x}}\,dx$

$$\int \frac{1}{\sqrt{5+4\cos x}}\,dx$$ I tried a lot to solve this simple looking problem. I tried $\cos x=2\cos ^2 {x\over 2}-1$ but no use. Tried half angle substitution of $\tan {x\over 2 }$ but ...
-1
votes
1answer
40 views

why is the following U-Substitution wrong?

It is known that $$ \int \frac{1}{\sqrt{1-x^2}} dx = arcsin(x)+c$$ this can be done utilizing u-substitution $ x = sin(u) $ However, i can let $ u = 1-x^2 $ $dx = -2u \, du $ which gives ...
0
votes
2answers
67 views

Integrating $\int\frac{\cosh(2y)\cos(2x)-1}{(\cosh(2y)-\cos(2x))^2}dy$

TL;DR: How does one work out the below integral (treating $x$ as a constant)? $$v=\int\frac{2\cosh(2y)\cos(2x)-2}{(\cosh(2y)-\cos(2x))^2}dy$$ Background: I need to work out the function $f(z)$, ...
0
votes
0answers
30 views

How to calculate this elliptic integral?

I need to calculate this integral $\int \cos(t\theta)\sqrt{1-k^{2}cos(\theta)^{2}} \mathrm{d}\theta$ But according to Wolfram|Alpha there is no result in terms of standart mathematical functions. ...
2
votes
1answer
39 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.
0
votes
1answer
47 views

Trigonometric Integration with $\sin(2x)$ and $\cos(2x)$

Finding value of $\displaystyle \int \frac{\cos(2x)\sin(4x)}{\cos^4(x)(1+\cos^2 (2x))}dx$ Try: let $$I =\int\frac{8\sin(2x)\cdot \cos^2(2x)}{(1+\cos 2x)^2\cdot (1+\cos^2(2x))}dx$$ Now put $\cos (2x)=...
2
votes
1answer
43 views

How first part of the Fundamental theorem of calculus works? [closed]

First fundamental theorem of calculus uses a function $F(x) =\ \int_a^xf\left(t\right)\,dt$ for f a continuous function between $[a,b]$ where $x$ is between $[a,b]$ $F(x)$ is an antiderivative of ...
0
votes
0answers
17 views

Indefinite integration of the binomial coefficient with variable second value

I can find $\int{\binom{x}{1}}dx$,$\int{\binom{x}{2}}dx$, etc. Using the definition that I know as $$\binom{n}{k} = \frac{n!}{(n-k)!k!}$$ I am trying to find $$\int{\binom{x}{n}}dx$$ in terms of both ...
0
votes
3answers
78 views

Evaluate the integral $\int \frac{\ln(x^2+1)dx}{x}$

I am interested in finding an exact expression for the integral $$\int \frac{\ln(x^2+1)dx}{x}$$ I start by using the transformation $w=\ln(x^2+1)$ leading to $e^{w}dw=2xdx$. Unfortunately, I couldn't ...
0
votes
4answers
40 views

Integrating a function with two variables

If $y=f(x)$ and $\dfrac{dy}{dx} = \dfrac{x}{\cos(y)+1}$ find $y$ given that $y$ passes through $(0,0)$. So we want to find $$\int \frac{x}{\cos y+1}dx$$
1
vote
2answers
48 views

What is the solution to the differential equation $\frac {dy}{dx} + xy = x$, when $y(0)=-6$? [closed]

I know about implicit differentiation and integrals, but how do I solve this type of equations?
1
vote
1answer
72 views

Unknown Indefinite Integral in my research

In my research appeared the following integral: $$\int e^{-a x^2 } \frac {b x^2}{c + k x} dx$$ Where $a$, $b$, $c$ and $k$ are constants. I know that the result of this indefinite integral is no ...
0
votes
3answers
122 views

Integrating a function with $\mathrm dx$ as an exponent [closed]

I am a high school student learning calculus. I encountered this what seems to be a challange problem:$$\int(x^{\mathrm dx}-1)$$We have learned some integrating techniques, but we still didn’t learn ...
1
vote
1answer
76 views

How to integrate $\arctan^2(x)$

I was wondering how you integrate $\arctan^2(x)$. I tried doing it by parts and allowing $u=\arctan(x)$ and $\frac{\mathrm dv}{\mathrm dx}=\arctan(x)$. But from then it becomes complicated, I was ...
0
votes
1answer
33 views

Applying Fourier transform to represent equation with integral as a sum of variables

According to a paper the equation with integral $\int_{-\infty}^{\infty}dx \rho_0(\lambda)e^{-b\lambda}=1/N$ (#1) where $\rho_0(\lambda)$ is a distribution function, $N$ is a natural number, can be ...
1
vote
0answers
47 views

Integral with Logs

I have a two integrals mathematica won't solve and I'm not sure how to solve them although I'm certain they are solvable. The first one is: $\int_{0}^\pi \log\left(A-B\cos\theta\right)\left(A+B\cos\...