Questions tagged [indefinite-integrals]

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

0
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3answers
83 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
50 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
56 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
58 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
66 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
112 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
85 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
118 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
67 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
74 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
28 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
4answers
87 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
73 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
34 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
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.
0
votes
1answer
48 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
18 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
79 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
45 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
50 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
126 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
38 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
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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\...
0
votes
1answer
66 views

Proving that $\cos(1/x)$ has a primitive

Let $f:\mathbb R\to\mathbb R$ with $$f(x)=\cos\left(\frac1x\right)\quad\text{for}\quad x\neq0,$$ $$f(x)=0\quad\text{for}\quad x=0.$$ I want to show that there exists a function $F:\mathbb R\to\mathbb ...
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0answers
19 views

Integration of a Product of a Complementary Error Function and Exponential function

I have derived a solution that describes heat diffusion in a linear model that is subjected to an initial temperature distribution. The resulting solution is obtained using the method of separation of ...
0
votes
1answer
35 views

Answer verifcation: $\int\sin(ax)\cos(ax)\,dx$ where “a” is a constant

I was currently doing a trigonometric substitution, and I noticed my answer is not on Wolfram Alpha Answers. This is the integral which I had to solve: $$\int\sin(ax)\cos(ax)\,dx \ \mathbf{\ \ \ \ ...
4
votes
1answer
70 views

CAS integrals of discontinuous functions

Background This post is motivated by my interest in the performance of symbolic integrators in computer algebra systems (CAS's), such as Mathematica (MMA). I've found that, when an integrand has ...
1
vote
4answers
62 views

Integrating $\int\tan^3(x)\,dx$ in two different ways gives two different answers

I was trying to find the antiderivative of a function $$\int \tan^3(x)\,dx$$ However, due to substitution differences, my book has a answer of $$\frac12\tan^2(x)+\ln(\cos x)+C$$ while I got an ...
1
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0answers
59 views

Integrating $\left|f(x)\right|$ by pulling out $\mathrm{sgn}(f(x))$ from the integral

I tried doing the following integral: $\int_{0}^{\pi/4}\sqrt{1-\sin2x}\mathrm dx$. Firstly I completed the square by rewriting $1$ as $\sin^2x+\cos^2x$ to get the integral revised to this form: $$I=\...
0
votes
4answers
58 views

What is the value of integral $\int \frac{{d}x}{\sqrt {x^2 + 9}}$

What is the value of integral $\int \frac{{d}x}{\sqrt {x^2 + 9}}$ The answer is $\sinh^{-1} (\frac{x}{3})$ I have tried solving it by putting $x = 3\tan \theta$, and got the answer $\ln |sec tan^{-1} ...
0
votes
2answers
40 views

Integral of irrational fraction to find right substitution [closed]

I have to calculate the following integral by using a substitution which result in no so long calculations. What would it be? $$\int \frac {\sqrt{x-1}-2} {((x-1)^{1/4}-1) \sqrt{x-1 }}\ \mathrm d x$$...
0
votes
2answers
44 views

Why does a constant in an integral only sometimes have an antiderivate

Summary Consider the following indefinite integrals: A) $$\int (2 + 3) dx = 5\ + C$$ B) $$\int (2 + 3x) dx = 2x + \frac{3x^2}{2}\ + C$$ As you can see, simply appending $x$ after $3$ also affects ...
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
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1answer
50 views

Find $\int \frac{dx}{(2+\sin x)^2}$: stuck at $\frac{1}{2}\int \frac{du}{u^2+u+1} - \frac{1}{2}\int \frac{udu}{(u^2+u+1)^2}$

Find $\int \frac{dx}{(2+\sin x)^2}$. $\tan \frac{x}{2}:=u$, $\sin x = \frac{2u}{1+u^2}$, $dx=\frac{2du}{1+u^2}$. $$\int \frac{\frac{2du}{1+u^2}}{(2+\frac{2u}{1+u^2})^2}$$ $$\frac{1}{2}\int \frac{\...
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votes
2answers
60 views

The Integral of $\int \sin(ax) \cos(ax) dx$

What is the integral of: $$I=\int \sin(ax) \cos(ax) dx$$ My approach is down below. I have attempted the problem and posted it as an answer. I did the problem using trigonometric substitution. $$u=...
2
votes
2answers
96 views

Integrating $\int \sin^3(a x)dx$

I have to integrate the following, where a is a constant: $$I=\int \sin^3(a x)dx$$ I did the following: $$u=ax$$ $$\frac{du}{a}=dx$$ Which gets me to this point: $$\frac{1}{a}\int(1-\cos^2(u))\sin ...
-4
votes
1answer
58 views

How to solve given integral? [closed]

I need it for my Fouier series's coefficent. $$ \int_{-π}^{π}\left| x\right| \cos{5x} \, dx $$
-2
votes
3answers
66 views

Use a valid integration method to calculate $\int_{\pi/2}^{3\pi/2}\frac{\sin x}{x}\,dx$

We know that $f'(x)$ = $\dfrac{\sin x}{x}$ and $f(\pi/2)=0$, $f(3\pi/2)=1 $ Use a valid integration method to evaluate : $$\int_{\pi/2}^{3\pi/2}f(x)dx$$ i think $\int_{\pi/2}^{3\pi/2}\frac{\sin x}{x}\...
0
votes
1answer
50 views

Finding General Formula for Coefficients of Partial Fractions

I am trying to evaluate the integral as written below and I've tried the following: $$\int\prod_{i=1}^{m}\dfrac{1}{x-i}\mathrm dx=\int\sum_{i=1}^{m}\dfrac{a_i}{x- i}\mathrm dx=\sum_{i=1}^{m}a_i\ln\...
0
votes
0answers
50 views

Does there exist a foolproof indefinite integration process?

I've been searching for this for sometime now & the best I've found are computer oriented algorithms (typically relying heavily on reference arrays & thousands of stimulus/action rules). ...
1
vote
3answers
59 views

$\int \frac{1}{{1-2x-x^2}} \, \mathrm{d}x $ substitution

I have this integral. $$\int \frac{1}{{1-2x-x^2}} \, \mathrm{d}x $$ But I am unable to do it right and I just don't know where is the problem in my steps. My steps: Complete the square $$\int \...
0
votes
0answers
43 views

Application of the fundamental theorem of calculus

Consider a function $\phi: \mathbb{R}^K \rightarrow (0,1]$. Suppose that the partial derivative $$ \frac{\partial \log(\phi(x))}{\partial x_1} $$ exists for every vector $x\in \mathbb{R}^K$, where $...
6
votes
4answers
788 views

How to quickly solve partial fractions equation?

Often I am dealing with an integral of let's say: $$\int\frac{dt}{(t-2)(t+3)}$$ or $$\int \frac{dt}{t(t-4)}$$ or to make this a more general case in which I am interested the most: $$\int \frac{...
0
votes
1answer
60 views

Why is my integration solution wrong?

This is my solution of this problem : Question. $\displaystyle \int \frac{x^3}{1+x^2} \, \mathrm{d}x$ Solution. Let $1+x^2 = u$. Then $$\frac{\mathrm{d}u}{\mathrm{d}x} = 2x \quad\...