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

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-1
votes
1answer
56 views

Compute $\int\frac{1}{3+\cos^3{x}}\mathrm{d} x$ [on hold]

I have an integral which seems hard for me: $$\int\frac{1}{3+\cos^3{x}}\,\mathrm{d}x.$$
0
votes
1answer
17 views

Definite Integals Obtained From Approximations Using Chebyshev Polynomial

Was trying to use chebyshev polynomial to obtain a cubic approximation to $f(x)=\frac{1}{x} $ I did it over the interval $[-1,1] $ Solving gave me the following four definite integrals: ...
5
votes
4answers
235 views

Problem of Integration by Parts involving algebraic and exponential functions

Can anyone please help me in solving this integration problem $\int \frac{e^x}{1+ x^2}dx \, $? Actually, I am getting stuck at one point while solving this problem via integration by parts.
0
votes
1answer
99 views

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $?

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $ ? I am completely at a loss , please help , thanks in advance .
1
vote
1answer
14 views

Question regarding elementary distribution theory

Let $D'(I)$ be the space of distributions on an open interval $I$, and let $D(I)$ be the space of test functions on $I$. I got the following homework assignment: "Define $u\in D'(\mathbb{R})$ be ...
4
votes
3answers
99 views

Why does WolframAlpha's expression for $\int\frac{dx}{x\sqrt{x^4-4}}$ disagree with my own?

$$\int\frac{1}{x\sqrt{x^4-4}}$$ My teacher gave us these notes and I'm unsure if they're correct. Wolfram gives a different answer, and when I derive I might have messed up. Thanks.
3
votes
0answers
65 views

Evaluate $\int \dfrac{1}{\sqrt{x-1}+\sqrt{x}+\sqrt{x+1}} \ \mathrm{d}x$ [duplicate]

Evaluate $$\int \dfrac{1}{\sqrt{x-1}+\sqrt{x}+\sqrt{x+1}} \ \mathrm{d}x$$ I tried rationalizing the denominator by twice multiplying, but it didn't do any good. I also tried trig ...
2
votes
0answers
53 views

How to evaluate the integral $\int\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$ [closed]

Please help me in doing this integration. $\int_{0}^{m}\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$ where m is a positive number.
0
votes
0answers
27 views

How does one integrate a function where the numerator is a polynomial of a degree n, and the denominator is a polynomial under root of degree m<n?

How does one integrate a function where the numerator is a polynomial of degree $n$, and the denominator is a polynomial under root of degree $m$ $(m<n)$? A random example being ...
2
votes
3answers
36 views

Evaluation of $\int\frac{2a\sin x+b\sin 2x}{(b+a\cos x)^3}dx$

Evaluation of $\displaystyle\int\frac{2a\sin x+b\sin 2x}{(b+a\cos x)^3}dx$ $\bf{My\; Try::}$Let $$\displaystyle I = \int\frac{2a\sin x+b\sin 2x}{(b+a\cos x)^3}dx = \int\left(\frac{a+b\cos x}{b+a\cos ...
2
votes
3answers
103 views

Primitive of $\int { \frac { x^{ 2 } }{ (x\sin x+\cos x)^{ 2 } } dx } $

How do I evaluate the integral of $$\int { \frac { x^{ 2 } }{ (x\sin x+\cos x)^{ 2 } } dx } $$ in a simple way? The way I could do the question, was by multiplying and dividing the fraction by $\cos ...
1
vote
2answers
20 views

Using partial fraction techniques to work out following intergrals

Hello I am studying for a mock test that is coming up, a question very similar to this one will be on this test, I have no idea how to complete this type of question, I have been given some vague ...
-5
votes
1answer
32 views

Antiderivation, application problem. [closed]

If the driver of a car want to increase the speed of 40 km / h to 100 km / h to travel a distance of 200 meters, what is the constant acceleration due stay? I only know that I need to use indefinite ...
12
votes
6answers
898 views

Where did I go wrong in my evaluation of the integral of cosine squared?

$$\int{\cos^2(x)}dx$$ Where did I go wrong in my evaluation of this integral? $$=x\cos^2x - \int-2x\sin(x)\cos(x)\,dx$$ $$=x\cos^2x + \int x\sin(2x)\,dx$$ $$=x\cos^2x + \left(\frac {-x\cos(2x)}2 ...
5
votes
3answers
155 views

Integrating and indefinite integral any possible way

How do I integrate the following: $$\large \displaystyle\int_0^{\infty}\frac{x^4e^x}{(e^x-1)^2} \, dx$$ I have tried everything from integrating by parts to simply expanding the denominator, but it ...
0
votes
2answers
52 views

Integrate the following problems

Let us consider the integral $$\int \frac{x^2}{(x\sin x+\cos x)^2}\, dx$$ I have tried with the following way $\displaystyle \int \frac{x^2}{(x\sin x+\cos x)^2}\, dx$ $\displaystyle \Rightarrow ...
0
votes
4answers
83 views

Integrating $\frac{1}{(x^4 -1)^2}$ [closed]

How to solve the the following integral? $$\int{\frac{1}{(x^4 -1)^2}}\, dx$$
3
votes
3answers
143 views

Evaluating $\int \frac{\sin\left(x\right)}{1+x^2}dx$

$$\int \frac{\sin\left(x\right)}{1+x^2}dx$$ I have tried to integrate by parts but it doesn't work. How do I evaluate it? Any advice, hint or well-thought solution will be appreciated.
1
vote
2answers
57 views

Why can I use Fubini' theorem on this function?

I used the fact that $\displaystyle \int_0^\infty\int_0^1 e^{-y}\sin(2xy)\,dxdy=\int_0^1\int_0^\infty e^{-y}\sin(2xy)\,dydx$ to solve $\displaystyle\int_0^\infty e^{-y}\frac{\sin^2(y)}{y}\,dy$. (The ...
6
votes
4answers
137 views

Integrating $\int \sqrt{x+\sqrt{x^2+1}}\,\mathrm{d}x$

Integrating $$\int \sqrt{x+\sqrt{x^2+1}}\,\mathrm{d}x$$ Using substitution of $x=\tan \theta$, I got the required answer. But is there a more elegant solution to the problem?
4
votes
5answers
172 views
+50

$\int\frac{1+x^2}{x^4+3x^3+3x^2-3x+1}dx$

$$\int\frac{1+x^2}{x^4+3x^3+3x^2-3x+1}dx$$ I tried to solve it. $$\int\frac{1+x^2}{x^4+3x^3+3x^2-3x+1}dx=\int\frac{1+x^2}{(x^2+1)^2+3x^3+x^2-3x+1}dx=\int\frac{1+x^2}{(x^2+1)^2+3x(x^2-1)+x^2+1}dx$$ ...
0
votes
3answers
62 views

Evaluation of $\int\frac{x^7+2}{\left(x^2+x+1\right)^2}dx$

Evaluation of $$\int\frac{x^7+2}{\left(x^2+x+1\right)^2}dx$$ $\bf{My\; Try::}$ Let $$\displaystyle \mathop{I = \int\frac{x^7+2}{(x^2+x+1)^2}}dx = \int\frac{(x^7-1)+3}{(x^2+x+1)^2}dx$$ ...
2
votes
3answers
37 views

Prove that $(2n+1)k_{n+1}=(2n+1)k_{n}+\cos^{2n+1} (x)$

Given that $$k_n=\int \frac{\cos^{2n} (x)}{\sin (x)} dx$$ Prove that $$(2n+1)k_{n+1}=(2n+1)k_{n}+\cos^{2n+1} (x)$$ I have tried to prove this is true by differentiating both sides with product rule: ...
0
votes
3answers
60 views

Integration with $\sin()$ and $\cos()$ [closed]

How do I solve this integration problem? $$ \int{\frac{1}{\sin^2(x)\cos^2(x)}\,dx} $$
3
votes
3answers
111 views

Find $\int \dfrac{\sqrt{1-x^2}}{1+x^2}\hspace{1mm}dx$

Find $$\int \dfrac{\sqrt{1-x^2}}{1+x^2}\hspace{1mm}dx$$ Any hints! I will do the work, just give me a clue
2
votes
3answers
105 views

Did Wolfram|Alpha mess up $\int 1 - \frac{1}{1-e^{-x}} \mathrm{d}x$ or did I?

I want to calculate $$\int 1 - \frac{1}{1-e^{-x}} \mathrm{d}x$$ Provided that $x>0$. Substituting $u=1-e^{-x}$, we get: $$\int \frac{1 - \frac{1}{u}}{1-u} \mathrm{d}u = -\int \frac{1 - ...
8
votes
3answers
125 views

Another integral $\int \frac{3 x^2+2 x+1}{ \left(x^3+x^2+x+2\right) \sqrt{1+\sqrt{x^3+x^2+x+2}}} \, dx$

Here is an indefinite integral that is similar to an integral I wanna propose for a contest. Apart from using CAS, do you see any very easy way of calculating it? $$\int \frac{1+2x +3 ...
-1
votes
0answers
21 views

Rewriting Integral Forms

I am asked to (1) Rewrite the integral $ \iint \limits _R f(x,y) \space \Bbb dx \Bbb dy$ in different coordinates $u,v$. (2) Hence, derive the form the integral $\iint \limits _R f(x,y) \space \Bbb ...
-4
votes
3answers
73 views

Need answer to a basic indefinite integral [closed]

Integrate $$\int \frac{x^2 + 3}{x^6(x^2 + 1)}\,\mathrm{d}x$$ with respect to x. I have no idea how to approach this problem. So it'll be better if you can provide the steps...
2
votes
6answers
71 views

Integration by Parts Question: Integrate $x^3e^x$

Evaluate $$\int x^3e^x \mathrm{d}x$$ I tried to use integration by parts to do this and I let $u = x^3$ and $\mathrm{d}v = e^x \mathrm{d}x$. So I get $$\int x^3e^x \mathrm{d}x= \int x^3e^x ...
3
votes
3answers
100 views

Finding $\int 2t\sqrt{4\sin^2(t) + 9\cos^2(t) + 4}\,dt$

I'm trying to find the result of the integral above. I have tried some simple substitution, and got it down to: $\int 2t\sqrt{8 + 5\cos^2(t)}\,dt$, which looks innocent enough. The solution is ...
3
votes
1answer
127 views

Integral of $2^{2^{2^x}}$?

$$\int2^{2^{2^x}}~\mathrm{d}x$$ Derivative is $\ln^3(2)2^{2^x+x+2^{2^x}}$. So no substitution technique can be used. So please guide, I am confused. Is this elliptic?
1
vote
2answers
67 views

Integral of $x e^{cx^3}$

How to evaluate the indefinite integral $\int x e^{cx^3}$. Is there any general form of solution for this integral? some function in terms of hypergeometric function or similar kind of functions? ...
3
votes
3answers
56 views

Find $\int \frac{5x^4+4x^5}{(x^5+x+1)^2}$

$\displaystyle \int \frac{5x^4+4x^5}{(x^5+x+1)^2}$ Since in the denominator of the integrand,$(x^5+x+1)^2$ is there.So the answer must be in the form $\displaystyle \frac{f(x)}{(x^5+x+1)}$, but i ...
1
vote
3answers
37 views

Integrating $\int\frac{xe^{2x}}{(1+2x)^2} dx$

I need help in integrating $$\int \frac{x e^{2x}}{(1+2x)^2} dx$$ I used integration by parts to where $u=xe^{2x}$ and $dv=\frac{1}{(1+2x)^2}$ to obtain ...
0
votes
1answer
70 views

Indefinite integration of $\sqrt{f(x)/x^2}$

If $$f(x) =\frac {x+2}{2x+3}$$ then the integral $$ \int{\sqrt{\frac{f(x)}{x^2}}}\,dx$$ is equal to? The expression is way beyond my comprehension, no idea in this case.
0
votes
0answers
43 views

Suggest a alternative method for the following indifinite integral. [duplicate]

The integral $$ \int{\sqrt{\tan x}}\ dx $$ can be solved by substituting $\tan x = t^2 $. But I am seeking an alternate way. Should I write $\tan x $ in terms of some other trigonometric function ?
-1
votes
1answer
46 views

Indefinite Integration [closed]

$$ \int \frac{dx}{\sin x + \sec x} $$ I tried substituting $\sin (x)$ as $t$ and was left with a tedious denominator. My friend tried to substitute $\tan \left(\dfrac{x}{2}\right)$ as $t$ but to no ...
5
votes
7answers
77 views

How to integrate $\int (x-1)\sqrt{x} \, \text{d}x$

How do I find this integral: $$\int (x-1)\sqrt{x} \, \text{d} x$$ I thought to use use substitution, but am not sure what I should use as $u$.
1
vote
1answer
82 views

What are the alternative methods for integrating the $f(x)=1/(\csc x + \cos x ) $?

What are the alternative methods for integrating this function? $$f(x)=\frac {1}{(\csc (x )+ \cos (x) )} $$ I found the answer by writing it as $sin (x)/(1+\sin (x)\cos (x))$ and using trigonometric ...
0
votes
0answers
41 views

Is there any shortcut method to check for correctness of an indefinite integral?

For the past few months I've been learning integral calculus.My experience is little and I have a (probably silly!) question. Sometimes computing indefinite integrals of certain functions takes a ...
2
votes
1answer
100 views

Integrating functions with $x^3$

After learning the integration of various functions with $x^2$ involved, I was given the following integration, as a challenge: $$\sqrt{1+x^3}$$ I tried various methods - too long to even try and ...
3
votes
5answers
92 views

Elegant solution for $\int {\frac{\cos(y)}{\sin^2(y)+\sin(y)-6}}dy$

I have the following integral: $\int {\frac{\cos(y)}{\sin^2(y)+\sin(y)-6}}dy$ I already know the solution, but it needs three substitutions. Is there a simpler, more elegant way to go about this?
3
votes
2answers
71 views

Help in finding the integral function.

Can somebody provide a hint in finding the following integral? $$\displaystyle \int \dfrac{1}{(x^3+1)^3} \text{ d}x$$ I thought of using partial fractions but that isn't making any sense.
6
votes
3answers
104 views

Integral problem

Find $$ \int e^{x \sin x+\cos x} \left(\frac{x^4\cos^3 x-x \sin x+\cos x}{x^2\cos^2 x}\right) \, dx$$ My attempt:I tried putting $x \sin x+\cos x=t$ and cannot express it in the form of $\int ...
3
votes
3answers
54 views

Finding indefinite integral.

I need hint in finding the integral of $$\int \frac{x^2}{(x \sin x + \cos x)^2} dx $$ I tried dividing the term by $x^2\cos^2x$ and then substituting $\tan x$.
3
votes
3answers
150 views

Help in finding the integral

I need help finding the indefinite integral of $$\int\,\frac{x}{(7x - 10 - {x^2})^{3/2}}\,\text{d}x\,.$$
-3
votes
3answers
69 views

How to calculate an elementary integral

How do you calculate $$\int\dfrac{2 du}{(u^2+1)^2}$$ It does not seem too difficult but I do not know which method to use.
-2
votes
1answer
35 views

Help with primitive function

I need help evaluating the indefinite integral $$\int\frac{\cos(5x) + \cos(4x)}{1-2\cos(3x)}dx.$$
4
votes
1answer
70 views

Calculation of integral using two different methods? [closed]

Find $$\int \dfrac{x^3}{(x^2+1)^3}dx$$ in two different ways, first using the substitution $u=x^2+1$ and then using the substitution $x=\tan \theta$. I managed to do both of these but the answer is ...