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

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2
votes
1answer
34 views

indefinite integrals equal imply integrands equal?

if there is an indefinite integral equality does it mean that there is an integrand equality? $$\int f(x)\,dx = \int h(x)\,dx \quad \overset{?}{\Longrightarrow} \quad f(x) = h(x)$$ I know that $$\int ...
5
votes
0answers
42 views

How to evaluate these indefinite integrals with $\sqrt{1+x^4}$?

These integrals are supposed to have an elementary closed form, but Mathematica only returns something in terms of elliptic integrals. I got them from the book Treatise on Integral Calculus by ...
0
votes
1answer
20 views

Integral of absolute value

I have the following integral which I want to make sure to solve correctly and transparently: \begin{equation} \int_{\mathbb{R}}\|e^{ax}\|dx \end{equation} If I take cases I obtain: ...
4
votes
0answers
73 views

How can we evaluate this tough integral?

$$ \int \frac{\sqrt{\sin\sqrt x}\cos \sqrt x}{1+x^2} dx $$ I have tried combinations of $x=t^2$, integration by parts, $\tan\left(\dfrac u2\right)$ substitutions it got even more complicated. Is ...
1
vote
2answers
48 views

Solve this indefinite integral, based on a volume problem

This is making me extremelly pissed off, because I saw a similliar integral that was apparently unsolvable, and now dear prof send this in the list without any resolution or help. The whole question ...
0
votes
4answers
71 views

Evaluating $\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$ using trigonometric substitution [on hold]

Using Substitution Integral Method, compute $$\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$$ (let $x=2\cos^2\theta+5\sin^2\theta$)
3
votes
3answers
232 views

Indefinite integral of a simple function

$\int 2(1 + \tan^2 x)$ My work : $2(1 + \tan^2(x) = 2 + 2\tan^2x$ $2x + \frac{2}{3}$ $\tan^3(x) \cdot \ln|sec(x)| + C$ The answer says no, after multiple tries :(
2
votes
4answers
56 views

Integration by parts of $\cos(x)e^{-x}dx$

I do the integral but I end up getting the original $\cos(x)e^{-x}dx$ on both sides and canceling them out resulting in no solution. Can I get a step by step break down of how to solve?
0
votes
1answer
33 views

Problem with this question on solid of revolution

Calculate the volume of a revolution solid obtained by rotation around the x-axis, the region bounded by the graph of $y=e^x$, $-1\le x \le1$ and the x-axis. Thanks in advance, and sorry about my ...
0
votes
1answer
18 views

About restricting variables in an integrand, and also changing the look of an integrands.

So, in the last step of, many, integrands, Wolfram chooses to restrict the $x$-values, even if I didn't specify which values $x$ can take on. Take for example: $$\int\frac{dx}{x(x^2-1)^{3/2}} = ...
3
votes
3answers
60 views

Problems with this integral $ \int \sqrt{1 + {1 \over t^2} + {2 \over t}} dt$

$$ \int \sqrt{1 + {1 \over t^2} + {2 \over t}}\,\mathrm dt$$ I tried making substitution, using $ u=1 + \dfrac{1}{ t^2} + \dfrac{2 }{ t} $, then , $dt=\dfrac{du}{-2\left({1 \over t^3 }+ {1 \over ...
4
votes
0answers
47 views

Can these two indefinite integrals be evaluated in closed form?

I'm wondering whether any of these two indefinite integrals $$\int \frac{1}{\sqrt{1+\alpha \sinh^{-4/3}(x)}}dx$$ $$\int \frac{\sinh^{-4/3}(x)}{\sqrt{1+\alpha \sinh^{-4/3}(x)}}dx$$ can be evaluated ...
0
votes
4answers
49 views

Integral with radical in denominator: $\int \frac{dx}{x(x^2-1)^{3/2}}$

I tried trigonometric substitution but it got me nowhere, and I can't find any examples online which has a radical in the denominator and a factor of $x$ outside of it. Own attempt: $$\int ...
8
votes
3answers
224 views

Indefinite Integral of Reciprocal of Trigonometric Functions

How to evaluate following integral $$\int \frac{\mathrm dx}{\sin^4x+\cos^4x\:+\sin^2(x) \cos^2(x)}$$ Can you please also give me the steps of solving it?
0
votes
1answer
49 views

Integration $\displaystyle\int \frac{x}{x^2-5x+6}dx$

Evaluate the Integral: $$\int \frac{x}{x^2-5x+6}dx$$ I solved twice and once I got $$3\log\left|x-3\right|-2\log\left|x-2\right|+C$$ and I tried again and changed one step and I got ...
1
vote
2answers
45 views

Evaluating $\displaystyle\int\frac{du}{\sqrt{-xu^{2}+yu+z}}$

This integral is just a step in a much longer problem for physics, but I am having some trouble with it. $$\int\frac{ \mathrm du}{\sqrt{-xu^{2}+yu+z}}$$ $x$, $y$ , and $z$ are constants Also ...
5
votes
2answers
76 views

Prove reduction formula for $\int \cos^n (x)\sin^m (x) \, dx$

$$\displaystyle\int \:\sin^n\left(x\right)\cos^m\left(x\right)\mathrm dx=\frac{\sin^{n+1}x\cos^{m-1}x}{m+n}+\frac{m-1}{m+n}\int \:\sin^nx\cos^{m-2}x\,\mathrm dx$$ I have been trying to solve for ...
-2
votes
1answer
61 views

solve the integral$\int\frac{x}{(\csc x)-x-x^2}dx$

$\int\dfrac{x}{(\csc x)-x-x^2}dx$ Its first time I solve same this integration I'm not sure what I can do. I had though the let $u=x~dx$ and $dv=\dfrac{dx}{(\csc x)-x-x^2}$ I can't solved it.
6
votes
2answers
128 views

Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?

My professor says that writing this is convenient $$\int \frac 1x \mathrm{d}x = \ln|x| + C\tag{1}$$ but wrong, since it should be written as: $$\int \frac 1x \mathrm{d}x = \begin{cases}\ln x + C ...
4
votes
2answers
84 views

How solve $\int \frac{dx}{(x^2-x)^x}$ [closed]

I want solve $$\int \frac{dx}{(x^2-x)^x}$$. thanks for help
1
vote
3answers
88 views

Evaluating $\displaystyle\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$

Evaluate the following limit: $$\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$$ Any advice on how to tackle this problem ?
2
votes
4answers
107 views

Evaluating $\displaystyle4\int \frac{\tan^2x\:\sec\:x}{\sec\:x\:+1}dx$

I was solving following integral $$\int \frac{\sqrt{x^2+4}}{\frac{x}{2}+1}dx$$ I think I need do a trigonometric substitution but I eventually end up with $$4\int ...
1
vote
1answer
68 views

Demostrate $\int \frac{dx}{(a\sin x+b\cos x)^{n}} = \frac{A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+c \int \frac{dx}{(a\sin x+b\cos x)^{n-2}}$ [closed]

Demonstrate: $$\int \frac{dx}{(a\sin x+b\cos x)^{n}} = \frac{A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+c \int \frac{dx}{(a\sin x+b\cos x)^{n-2}}$$ $A,B$ are undetermined coefficients
2
votes
3answers
103 views

Where did I go wrong in this integration $\int\frac{\ln(1-e^x)}{e^{2x}}\,dx$

Here is the closest I've come to the answer Link to Wolfram equality not giving true as output NB! I marked where I'm unsure in $\color{red}{red}$ color. And please don't get startled because of the ...
0
votes
3answers
57 views

Evaluate $\displaystyle\int _{-1}^0\:\frac{\left(x+6\right)}{x^2+2x+2}\:dx$

Evaluate following integral: $$\int _{-1}^0\:\frac{\left(x+6\right)}{x^2+2x+2}\:dx$$ I tried to solve it but don't know how to do it, can anyone please help
5
votes
2answers
127 views

Evaluation of $-\int e^{\cos(t)}\sin(\sin(t)+t)\,dt $

How would I integrate this: $$-\int e^{\cos(t)}\sin(\sin(t)+t)\,dt $$ I have tried several methods but can't seem to work this out.
-2
votes
3answers
43 views

Evaluating $ \int\frac{x}{\sqrt{3-x^2-2kx}}\,dx $ [closed]

I'm trying to evaluate this integral: $$ \int\frac{x}{\sqrt{3-x^2-2kx}}\,dx $$ where $k$ is a real parameter.
2
votes
4answers
103 views

Indefinite integral of $\frac{\sqrt{x}}{\sqrt{x}+1}$

For this I tried using the substitution technique, but it got me nowhere near the right answer. What my notepad looks like: $$f(x) = \dfrac{\sqrt{x}}{\sqrt{x}+1}$$ and $$F(x) = \int f(x) = ...
1
vote
0answers
61 views

Problem in understanding the process of finding antiderivative.

Antiderivative or indefinite integral is the family of functions the derivative of which gives the original function. Now, let's elaborate the process. Suppose $F(x)$ is the derivative of the ...
-1
votes
1answer
22 views

Evaluate the integral (using partial fractions maybe?) [duplicate]

Evaluate the following integral $\int{\frac{1}{(x+a)(x+b)}}$ (this might involve partial fraction decomposition, $\int{\frac{1}{x^2+x(a+b)+ab}}$ this is what my first step was)
-4
votes
2answers
105 views

Evaluating $\int 2x e^{x^2} \, dx$ [closed]

Can I have some help or pointers on how I should evaluate following indefinite integral? $$\int 2x e^{x^2} \, dx$$
0
votes
1answer
30 views

Finding $f(x)$ from first and second derivitive

What is f(x) when $$f(1)=0$$ first derivative $$f(1)=1/2$$ second derivative $$f(x)=1/x^3$$ Currently i have tried where the second derivitive = first derivitive + constant at x=1 1/x^3 = f^i(x) + ...
5
votes
3answers
103 views

Integration of $1/(x^4 \sin x +2x)$

$$\int\frac{1}{x^4 \sin x +2x} dx\ $$ How to evaluate this integral. How to go about evaluating these integrals?
0
votes
0answers
18 views

The Area of Right Triangle and Integral

Consider that we are dealing with a right triangle with constant base ($B=B_1$ and $\frac{dB}{dt}=0$). The values (of $B=B_1$ and of $\frac{dB}{dt}=0$) are maintained by nature to be constant from ...
1
vote
5answers
121 views

What is indefinite integral actually - $\int f(x)dx$ or $\int_a^x f(t)dt$?

What is indefinite integral? This is the question that always perplexes me. First my book wrote that Indefinite integral of $f(x)$ is $F(x)$ if on differentiation, it gives $f(x)$. In fact it is ...
-1
votes
0answers
23 views

The Area of Right Triangles and Fundamental Theorem of Calculus

Consider that we are given a right triangle with constant base $B=B_1$ and $\frac{dB}{dt}=0$. The values (of $B=B_1$ and of $\frac{dB}{dt}=0$) are maintained by nature to be constant from the ...
3
votes
3answers
53 views

Exercise with $u$-substitution

How would I "see" this or go a head to solve it? I just can't understand it or see why I would go that way; how can I make myself able to see stuff like this? $$\int \frac{t}{ \sqrt{4-t^4}}dt $$ ...
-2
votes
1answer
47 views

Integrability of a Dirichlet Function

Let $f:[0,1]\to \mathbb{R}$ $$ f(x)=\begin{cases}x & x\ \mbox{is rational} \\ -x & x\ \mbox{is irrational} \end{cases} $$ Prove that the function $f$ is not integrable. Use darboux sums and ...
2
votes
2answers
31 views

Calculate the following intergral

I have to calculate the following integral $$ \int \sqrt[3]{1+x\ln{x}} * (1+\ln{x}) dx$$ I have thought about using the following notation: $$ t = {1+x\ln(x)} => x\ln{x} = t-1 $$ But here I ...
6
votes
2answers
92 views

Evaluating $ \displaystyle\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$

Evaluate $$\displaystyle \int e^{x\sin x+\cos x}\left(\frac{x^4\cos^3 x-x\sin x+\cos x}{x^2\cos^2 x}\right)dx$$ $\bf{My\; Try::}$ Let $$\begin{align}I &= \int e^{x\sin x+\cos ...
1
vote
1answer
38 views

Are there examples of nontrivial $f$ for which the antiderivative of $\tan\circ f$ is known?

I'm looking for an example of a function $f$ (apart from inverse trigonometric and linear functions) such that $\int\tan(f(x))dx$ is known. Special functions included in the typical CAS are acceptable ...
3
votes
2answers
48 views

Evaluating $\int (\sin x)^2\,dx$ [closed]

Evaluate $$\int (\sin x)^2\,dx$$ Anyone can guide me for this? Thank you!
1
vote
1answer
38 views

$\int \tan(x)\operatorname{tanh}(x), \operatorname{dx}$

I was wondering if there existed a closed form for $$\int \tan(x)\operatorname{tanh}(x), \operatorname{dx}$$ I don't think this integral has a closed form, but could it be evaluated over some points ...
0
votes
1answer
22 views

cubic integral roots

I am trying to find the integral roots (if they exist) of the following polynomial. Additionally, it would be helpful if someone could explain an algorithmic approach to solving this. $$ f(x) = 2x^3 ...
1
vote
3answers
66 views

Integral of $\frac{x}{\sqrt{1+x^5}}$

I am trying to calculate the following integral: $\displaystyle\int_0^\infty \frac{x}{\sqrt{1+x^5}}\, dx$ But I can't seem to find a primitive for that function. I was trying to find a good ...
3
votes
1answer
99 views

Evaluation of the integral $\int \sqrt{t^4-t^2 + 1}\,dt$

My friend took his Calculus $2/3$ test yesterday. One of the questions he had trouble with was this integral: $$\int \sqrt{t^4-t^2 + 1}dt$$ My attempt It seems rather clear that the only approach ...
1
vote
1answer
13 views

Simplifying an indefinite integral representing the calculation of an average

I have an equation for a term $z_i$: $$ z_i = \ln\frac{a_iR+p_i}{T_o* tan\theta_i } $$ This represents a value in a grid, at location $i$, with the grid representing a geographic area. To get the ...
0
votes
1answer
20 views

Integral of $ye^{-(x+1)y}$

Not sure where I'm going wrong on this one. $$\int{ye^{-(x+1)y}}\:dy$$ $$u = y \qquad du = dy$$ $$dv = e^{-(x+1)y} \qquad v = -\frac{e^{-(x+1)y}}{x + 1}$$ $$-\frac{ye^{-(x+1)y}}{x + 1} \times ...
0
votes
2answers
66 views

Explain each step to find $ \int \frac{\cos x\,d x}{1 + (\sin x)^{2}} $

I know the answer but I don't understand the steps to integrate. $$ \int \frac{\cos x\,d x}{1 + (\sin x)^{2}} $$
1
vote
3answers
51 views

Indefinite integral of $\frac{\arctan x}{x^2+1}$

EDIT: I was studying from a site that uses really ambiguous notation so I misread $\arctan\ (x)^2$ as $\arctan\ (x^2)$. Now I can see why the integral is actually $\frac{1}{2} \arctan^2\ x + c $. ...