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Questions tagged [indefinite-integrals]

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

0
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1answer
47 views

Integral of $\int{\frac{1}{\sqrt{x(1+x^2)}}dx}$

I was trying to solve the following question: Evaluate: $$\int{\frac{1}{\sqrt{x(1+x^2)}}dx}$$ This is an unsolved question in my sample papers book and so I believe it should have an elementary ...
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0answers
48 views

How to calculate the integral $\int\frac{e^{ixy}}{w^2-x^2}dx$?

How to calculate the integral$$\int_{-\infty}^{\infty}\frac{e^{ixy}}{w^2-x^2}dx$$ where w,y are constants. I tried to separate them as $$\frac{1}{2w}\int_{-\infty}^{\infty}(\frac{e^{ixy}}{w+x}+\...
1
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0answers
16 views

Products of logarithms under the integral sign

This question is a simplification of a previously asked question: Polylogarithmic integrals Consider the following type of function: \begin{equation} \int \frac{\prod_{i=1}^N \log(x-\beta_i)}{x-\...
3
votes
2answers
70 views

Integrating $\int_{0}^{a}\sqrt{{\tanh}(a)-{\tanh}(x)}\;dx$

What is the integral $$\int_{0}^{a}\sqrt{{\tanh}(a)-{\tanh}(x)}\;dx$$ I have tried various substitutions but couldn't get the answer. The substitution $x = {\tanh}^{-1}y$ has simplified into ...
2
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1answer
28 views

A quick question on in logs

I was solving indefinite integrals $$\int_0^22^x x \,dx$$ I use ILATE as a clue to consider the first function and second function. $2^x$ is a algebraic function or logarithmic function?
1
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3answers
50 views

Integration by substitution to take out square root

Find $$\int (x+1)\sqrt{x^2+1}\,dx .$$ In order to not bother with the square root I thought of doing this: $let$ $ x^2+1=(x+t)^2$ $\Rightarrow$ $x=\frac{1-t^2}{2t}$ $\Rightarrow x-t=\frac{1+t^2}{2t}$...
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5answers
47 views

Any thoughts on this integral?

$\int \cos^2(x)\cdot\sin^4(x)dx$ I tried the usual trigonometric identities but they don't seem helpful
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6answers
78 views

What's the answer to $\int \frac{\cos^2x \sin x}{\sin x - \cos x} dx$?

I tried solving the integral $$\int \frac{\cos^2x \sin x}{\sin x - \cos x}\, dx$$ the following ways: Expressing each function in the form of $\tan \left(\frac{x}{2}\right)$, $\cos \left(\frac{x}{2}\...
2
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3answers
52 views

Why can't I make the substitution $ u = \sin (ax + b) $ to evaluate $ \int \sin (ax + b) \cos (ax + b) dx$?

Evaluate $ \int \sin (ax + b) \cos (ax + b) dx$? To do this, I started of by substituting $ u = \sin (ax + b) $. That made $ du = cos (ax + b) \cdot a $ and wrote the integral as $ \frac 1a \int u \ ...
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0answers
32 views

calculate this triple integral with $f(x,y,z)=(x^2+y^2)e^z$

I need to evaluate on this region : $D=\{1\le z \le 2,x^2+y^2 \le z^2 \}$ on this function : $f(x,y,z)=(x^2+y^2)e^z$ So the triple integral should be : $\int_{0}^{2\pi}\int_{1}^{2}\int_{0}^{z}r^3e^...
1
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1answer
22 views

Calculate flux Triple Integral

$R=\{z^2-4z+y^2 \le 0,0 \le x \le 1\}$ with $F=(x\sqrt{y^2+z^2},-z,y)$ So it's a shifted cylinder : $(z-2)^2+y^2=4$ $$ \left\{ \begin{aligned} x&=x\\ y&=2\sin\theta\\ z&=2+2\cos\theta \...
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1answer
56 views

calculate this line integral

this is my curve : $r(t)=(\cos{t},\sin{t}-1,2\cos{\frac{t}{2}})$ , $t=[0,3\pi]$ $r'(t)=(-\sin{t},\cos{t},-\sin{t})$ $||r'(t)||=\sqrt{\sin^2{t}+1}$ I have to calculate: $\int{(y+1)}ds$ So I have : ...
3
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3answers
77 views

How do you integrate $\int \frac{\cos(4x)}{\cos(x)}dx$?

I tried using trigonometric formulas for turning it into 2$\int \frac{\cos^2(2x)}{\cos(x)}dx - \int \frac{1}{\cos(x)}dx$ and can solve the second one, but still no idea of how to proceed with $\int \...
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3answers
70 views

integrating by parts $9e^{2x}\cos(3x)$

integrating by parts: $$\int 9e^{2x}\cos(3x)dx$$ It seems like whichever part I start with integrating or deriving It still leads to a "by parts" integral. How Do I deal with it?
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1answer
59 views

integrating $\frac{8\sin(2\ln x)}{\frac 2x}$ [closed]

How do I integrate: $$\int\frac{8\sin({2\ln x})}{\frac 2x}\ dx\,?$$ I'm not able to do it by parts...
2
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2answers
59 views

Primitive of a function with $\sin \frac{1}{x}$

I have the next integral: $$\int\biggl({\frac{\sin \frac{1}{x}}{x^2\sqrt[]{(4+3 \sin\frac{2}{x})}}}\biggr)\,dx ,\;x\in \Bigl(0,\infty\Bigr)$$ I used the substitution $u=\frac{1}{x}$ and I got $$-\int\...
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0answers
66 views

Proving the answer to the integral of $\sin(π/x^2)$

When I integrate $\sin(π/x^2)$ on W|A, I get: \begin{equation} \int \sin\left(\frac{π}{x^2} \right)\:dx = x\sin\left(\frac{\pi}{x^2}\right) - \sqrt{2}\pi C\left( \frac{\sqrt{2}}{x} \right) + D \end{...
8
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3answers
1k views

How can I calculate $\int\frac{x-2}{-x^2+2x-5}dx$?

I'm completely stuck on solving this indefinite integral: $$\int\frac{x-2}{-x^2+2x-5}dx$$ By completing the square in the denominator and separating the original into two integrals, I get: $$-\int\...
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0answers
39 views

Indefinite integral of $\frac{1}{\sin(\ln x)}$

I have to find the indefinite integral of $\frac{1}{\sin(\ln x)}$ and it seems that it doesn't work with the method used for the integral $\sin(\ln x)$. Is there anyone who can help me with this? Your ...
4
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0answers
70 views

Hypergeometric identity

I was trying to solve this integral problem and I noticed something that may be true $$ \int((1-x^r)^{1/r}-x)^{2 n} \mathrm dx = \frac{1}{2 n+1}\sum _{j=1}^{2 n+1} (-1)^{j+1} x^j \binom{2 n+1}{j} \, ...
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1answer
57 views

Solving an unsolvable integral ?? [duplicate]

I recently stumbled upon an indefinite integral . sin(x)/x [ Another similar one is root (x) times sin x. However if we substitute sin(x) in terms of x as Maclaurin series we could get a series of ...
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1answer
38 views

The integral of

I have to take the indefinite integral of the following function: $$\int_\limits{0}^{\frac{\pi}{2}}\sin^2(\frac{1}{3}\theta)d\theta$$ I did a double substitution, my first was: $$3\int_\limits{0}^\...
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4answers
69 views

Integral utter confusion with substition and dx/du

I need to find the indefinite integral I = $$\int e^x (1+e^x)^{\frac{1}{2}}$$ by using a proper substition method. I tried it on https://www.integral-calculator.com and it gave the following ...
2
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1answer
105 views

Integral involving hypergeometric function

I've worked out the projection of a spherically symmetric power law volume density profile $\rho(r)=br^a$, i.e. its surface density $\sigma(R)$, and am now trying to integrate this in a series of ...
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0answers
68 views

Integrate $x\operatorname{erf}^{\,3}(x)\,e^{-x^2}\,dx$

Looking for a way to perform this integral related to the error function. I am thinking an answer in closed form cannot be done, but hoping I missed something. $$ \int x\operatorname{erf}^{\,3}(x)\,e^...
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0answers
60 views

Introducing a new variable for evaluating integrals

In some problems we use reverse substitution for evaluating integrals . For example consider $\int \sqrt{1-x^2}dx$ and $x = \sin(t) \ , \ \frac{-\pi}{2}\le t \le \frac{\pi}{2}$ . In this case , ...
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2answers
64 views

Not getting the right answer with alternate completing the square method on $\int\frac{x^2}{\sqrt{3+4x-4x^2}^3}dx$

So I've looked up how to do this problem and when they complete the square it's using the $(\frac{-b}{a})^2$ method where, in the denominator, they factor out the $4$, and then make: $$x^2-x=x^2-x+{1\...
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0answers
95 views

Why is there no antiderivative to the integral of sin(x)/x [duplicate]

I have been baffled for a while now because no antiderivative has been found for what seems to be a simple enough Integral on the surface. $$\text{Si}(x)= \int \frac{\sin(x)}{x} dx $$ Is this due ...
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0answers
16 views

The function defined by the sum of non-decreasing absolutely continuous functions on $\mathbb{R}$ is continuous and differentiable a.e.

True or False? If $\{f_n\}_{n\ge 0}$ is a sequence of non-decreasing absolutely continuous functions on $\mathbb{R}$, and if $f:=\sum_{n\ge0}f_n(x), x\in\mathbb{R},$ is finite at each $x$, then $f$ is ...
2
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3answers
41 views

Cancelling with absolute values

I am trying to understand how cancellation works in integrals when there are absolute value expressions involved. For example: $$\int\frac{sinx}{\sqrt{1-cos^2x}}dx = \int \frac{sinx}{\lvert sin(x) \...
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1answer
56 views

Solve the differential equation very tough to solve!!

Today a classmate of mine had given me and my teacher a differential equation I don't know from where but it became a headache for me and my teacher because we both were unable to solve it at some ...
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2answers
48 views

When finding the derivative of the integral of a function I get different results

I am using wolframalpha.com and other online calculus calculators. The problem I am solving: $$\int x\cos^2\left(x^2\right)dx$$ But the integral and its derivative don't match. Am I doing something ...
2
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4answers
92 views

Solving $\int \frac1{\cos x}\mathrm dx$ [duplicate]

I'm trying to solve the following integral: $$\int \frac1{\cos x}\mathrm dx$$ I know that $\int \cos x\mathrm dx$ = sin x, but I don't know how to proceed with $\frac1{\cos x}$.
2
votes
3answers
109 views

How do I integrate the function $\sqrt{(6x + 2)}$?

How do you integrate $\sqrt{(6x + 2)}$? I've tried to use the following substitutions: let $x = \sin(u)$ and $dx = \cos(u)$ (along the lines of the Yahoo Answers link). I tried looking for simple ...
2
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2answers
63 views

Integral of $\sec(x)$ using $u$ sub

I've just begun learning how to integrate and I wanted to see if I could integrate $\sec(x)$ by $u$-substitution. After getting my answer, I was told it couldn't be in complex form, but why, and if so,...
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1answer
62 views

I am trying to change the order of integration of a problem and I am getting confused at a point

My math book has a question of changing order of the following integral: $$\int_0^a\int_{\sqrt{ax-x^2}}^{\sqrt{ax}} f(x,y)\,dx\,dy.$$ So basically I have to change $dxdy$ into $dydx$ and their ...
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3answers
70 views

Integration of powers of trigonometric function with linear term

I got stuck trying to find a general formula for the following integral $$\int_0^{\pi} t \cdot\cos^{2n}{\left(\frac{t}{2}\right)} \, dt = 4 \int_0^{\pi/2} t \cdot\cos^{2n}t \,dt \; , \; \text{ for } ...
1
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1answer
82 views

$ \int \bigl| a\sin(x)+b\cos(x) \bigr| {\rm d}x = ? $

My friend evaluated this to be $$ \int \bigl| a\sin(x)+b\cos(x) \bigr| {\rm d}x \\ = \sqrt{a^2+b^2} \left( \sin(x-\phi)\text{sign}(\cos(x-\phi)) +\frac{2}{\pi} \bigl(x-\arctan(\tan(x-\phi)) \bigr) \...
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0answers
51 views

Fractional/Integer Based integrals

I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?
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1answer
78 views

What does the function $E$ stand for in WolframAlpha's solution to this integral?

While trying to find the circumference of an ellipse, I came up with this result in Wolfram Alpha. Equation: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} =1$ While trying to perform the definite integral, ...
3
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2answers
328 views

Can an indefinite integral be expressed as a definite integral with variable bounds?

If I have a function $f(t)$, and an indefinite integral of this function, $g(x) = \int f(t)\, dt$, is there any way I can express $g(x)$ as a definite integral whose bounds depend on $x$? I thought I ...
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0answers
20 views

Indefinite integration of a function with a trigonometric function raised to a high power [duplicate]

I need to integrate indefinitely the expression $$ \frac{\csc^{2}x - 2017}{\cos^{2017}x} $$ I am unable to handle this high power of $\cos(x)$ . A hint would be useful for me.
3
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1answer
70 views

Integral $\int \frac{\sin^n(x)}{\cos(x)}dx$

In one of my exercises about integration we had to solve the following integral: \begin{equation} \int \frac{\sin^n(x)}{\cos^m(x)}dx \end{equation} We had to do this via a recursive integral. I ...
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3answers
124 views

integrating $\int \frac{1}{e^x +1}\:dx$ [duplicate]

I've found a method for integration of $\frac{1}{e^x +1}dx$. However none of the information I found explains the intuition. It might be clear for most people, but I was hoping that somebody could ...
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1answer
50 views

Indefinite integral of reciprocal n degree polynomial

I find that the Torricelli's trumpet has very interesting properties. It's surface: $$\int_{1}^{\infty}\frac{2\pi}{x}dx=2\pi\ln{x}|_{1}^{\infty}\rightarrow\infty.$$ is infinity. Its volume: $$\...
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1answer
29 views

Substitution for integrals two different cases.

In my book they make the point that: "The method of substitution cannot be force to work". They then go on to say that: "there is no substitution that will do much good with the integral $\int x(2+x^...
1
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1answer
49 views

Velocity function of a physical scenario

This may belong in the physics stack exchange, but please hear me out and it does involve mathematics. Two planets, both of mass m, are separated by a distance x (that is from one planet's center to ...
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0answers
40 views

How can I prove the following integral identity?

The following equation is a solution of a convection-diffusion problem. I want to prove that the resultant solution is simplified to the initial condition when $\Delta t$ approaches zero. Consequently,...
1
vote
1answer
49 views

How can this integral be solved? Is is an indefinite integral?

The integral I want to solve is $$\int_1^2 \frac {2 \ln(x)}{x+1} dx$$ I tried to integrate it by parts in 2 ways and I tried to do the integral by parts twice, I thought of a change of variable and ...
-1
votes
2answers
98 views

trigonometric integration problem [closed]

Hello guys can someone please help me find the answer : $$\int \sin x \tan x~dx$$