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

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-2
votes
2answers
71 views

How to evaluate $\int \frac{\mathrm dx}{1+\sin x−\cos x} $?

Is there someone show me how I evaluate this integral:$$\int\frac{\mathrm{d}x}{1+\sin x−\cos x} $$ I used $t=\tan\frac{x}{2}$ but i didn't succeed . Thank you for any help .
3
votes
4answers
145 views

Is integration of $x\operatorname{cosec}(x)$ defined?

Is integration of $x\operatorname{cosec}(x)$ possible? If yes, then what is its closed form; if not, then why is it non-integrable ?
-2
votes
3answers
67 views

Integral of rational function with a squared term in the denominator

I know the integration when in the reciprocal there's only degree $1$, but what about degree $2$? Take an example, $$\int\frac{x \, \mathrm{d}x}{a+bx^2}$$
1
vote
0answers
41 views

An intriguing integral $\int \ln{|\nabla u|^2}\, \mathrm{d}u$

How to integrate $\int \ln{|\nabla u|^2}\, \mathrm{d}u$ ? In Cartesian co-ordinates this would be $\int \ln{(u_x^2 + u_y^2)}\, \mathrm{d}u$, where $u_x \equiv \frac{\partial u}{\partial x}$. We ...
2
votes
2answers
37 views

How to solve the integral $\int\tan^{3}x \sec^{3/2}x\; dx$?

How to solve the following indefinite integral $$\int \tan^{3}x \sec^{3/2}x \; dx$$ to get the solution in the form of $$\large\frac{2}{7}\sec^{7/2}x - \frac{2}{3}\sec^{3/2}x +c$$ I tried taking ...
0
votes
2answers
66 views

How to integrate $\int\frac{3x+2}{x^2-x-2}dx$

This is the indefinite integral I have to evaluate: $$\int\frac{x^3}{x^2-x-2}dx$$ so by using the long division on polynomials technique, I got to: $$\frac{x^2}{2}+x+\int\frac{3x+2}{x^2-x-2}dx$$ How ...
-6
votes
0answers
54 views

Can any one help me solve this integral ??? [on hold]

![i cannot able to solve this integral ,can any one able to solve this integral and i used integral technique but i cannot able to solve this equation the integral is with respect to x ...
0
votes
1answer
49 views

Integrate the function by substitution method.

$$\int \frac1{ \cos(x-a)\cos(x-b)} \, \mathrm{d}x$$ Can someone help me to integrate this function by method of substitution.I am not able to start it for possibilities are not coming in my mind. ...
0
votes
2answers
60 views

A general method for integration of rational function.

$\int\frac {x^3}{1+x^5}$ ATTEMPT: I did the following substitution: Let $x=\frac{1}{t}.$ $dx=\frac{-1}{t^2}dt.$ substituting back: $I=\int\frac{-1}{1+t^5}dt$ which doesn't seems a simpler ...
5
votes
2answers
206 views

Evaluating the indefinite integral $\int\log\!\left(x+\sqrt{x^2-1}\right)\!dx$

I came across the following integral, and I don't know how to solve it. $$ \int\log\left(x+\sqrt{x^2-1}\right)dx $$ I tried the "obvious" substitution of $x=\sec\theta$, which gives you: $$ ...
1
vote
2answers
32 views

Finding the integral of rational function of sines

$$ \int \frac{\sin x}{1+\sin x} \, \mathrm{d}x$$ How do I integrate this? I tried multiplying and dividing by $ (1- \sin x) $.
0
votes
2answers
71 views

Integral of $x/(2x-1)$

I'm not sure how to do this, I'm also new to math.stackexchange so please excuse any novice mistakes. So anyways, here is a question I have on a summer assignment for Calculus BC (this is review from ...
2
votes
0answers
29 views

What is the example of a continuous function which is not an indefinite integral?

By Royden's Real Analysis (3rd), chap 5. Thm 14, it reads that 'A function F is an indefinite integral if and only if it is absolutely continuous. ' What is the continuous but ...
1
vote
3answers
52 views

How do i find this : $\int \frac{1}{(x+a) \sqrt{x+b}}\ dx$, where $a > b > 0$?

Is there someone show me how do I find : $$\int \frac{1}{(x+a) \sqrt{x+b}}\ dx$$, where $$a > b > 0$$ ? I tried to make it as sum of fraction to be easier but sorry i didn't up Thank you for ...
3
votes
7answers
133 views

Evaluating the indefinite integral $\int e^{-x^2}\,\mathrm{d}x$ [closed]

In my book, it is said that $$\int e^{-x^2} \, \mathrm{d}x$$ cannot be solved by the method of inspection. It then turned to method of substitution as a new topic. I am not able to solve this ...
-1
votes
0answers
46 views

Find indefinite integral $dx/(x^6+1)$. [closed]

Help to find indefinite integral $$ \int \frac{\mathrm{d}x}{x^6+1} $$
1
vote
2answers
68 views

Integral of $x^2\sqrt{5+x}\ dx$

I have the following integral to solve, with my working out below. This is a bit more complicated than I am used to, so I'm hoping for some feedback as I'm not sure if my process & solution are ...
-1
votes
0answers
41 views

How to find the following definite and indefinite integrals [duplicate]

I want to calculate the integral $$\int_0^{\pi \over 2}e^{ \sin t}dt$$ can we find a primary function for $f(t) = e^{\sin t}.$ With many thanks.
1
vote
2answers
42 views

Finding the following definite and indefinite integrals

I want to calculate the integral $$\int_0^{\frac{\pi}{2}} e ^{ \sin t}\, dt.$$ Can we find a primitive function for $f(t) = e ^{\sin t}$?
4
votes
1answer
67 views

Find all functions such that $\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$

Is it possible to find all functions such that $$\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$$? My teacher asked us to give examples to prove that this is not true but I was ...
13
votes
1answer
208 views

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion [duplicate]

(Note: I apreciate very much who marked this as a duplicate but I would like an answer for why my proof is wrong) This is my solution, I have no clue why it failed. Let's start: define $$I_n(m) = ...
1
vote
3answers
148 views

Find $\int_a^b \sin |x| \, \mathrm{d}x $

How to find the integral $$\int_a^b \sin |x| \, \mathrm{d}x \,?$$ I'm able to obtain definite integral of form $ \int_a^b \lvert\sin x \rvert \, \mathrm{d}x$ but not when the modulus operator is ...
0
votes
2answers
73 views

Integration problem: $\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $

I need help in solving the following problem: $$\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $$ I really don't know how to start solving this problem; any tips or solutions will be greatly ...
1
vote
2answers
88 views

Evaluate the indefinite integral $\int \frac{\cos \theta}{ \sqrt{2 - 9 \sin^2 \!\theta}} \mathrm{d}\theta$

I want to evaluate $$\int \dfrac{\cos \theta \, \mathrm{d}\theta}{ \sqrt{2 - 9 \sin^2 \theta}}$$ but I can't seem to get the answer, my working is as below:
1
vote
1answer
80 views

Need hint to solve a nasty integral.

Let $f(x)=\frac{x+2}{2x+3}$, $x>0$. If $$\int \left( \frac{f(x)}{x^2} \right)^{1/2}dx=\frac{1}{\sqrt{2}}g \left(\frac{1+\sqrt{2f(x)}}{ 1-\sqrt{2f(x)}} \right) -\sqrt{\frac{2}{3}}h ...
1
vote
4answers
97 views

Solving $\int \frac{dx}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$

This was in an old exam in a physics for mathematicians class. I haven't had to deal with these kind of integrals for a while and can't think of a decent substitution. I asked my teacher about it and ...
0
votes
1answer
85 views

How do I integrate $\frac{\sin x+\cos x}{\sin^4 x+\cos^4 x}$ [duplicate]

How do I integrate $$\frac{\sin x+\cos x}{\sin^4 x+\cos^4 x}$$ ? Tried different ways including the tangent half-angle substitution (which seems to be disastrous).
2
votes
1answer
52 views

Integral with complex variable

I want to compute $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{x+yi +2}} dy $$ where $i$ is the imaginary number. How to compute this integral??
2
votes
2answers
42 views

Help with indefinite integration

I am learning indefinite integration, yet am having problems understanding and recognizing where to substitute what. a good trick is to attempt convert algebraic expressions into trigonometric and ...
1
vote
1answer
53 views

If $k$ is a non-zero constant, determine by inspection the indefinite integral of $\int e^{kx} dx$.

I have to solve this exercise: If $k$ is a non-zero constant, determine by inspection the indefinite integral of $\int e^{kx} dx$. By inspection, I guess it means that it should be solved by ...
3
votes
1answer
50 views

Evaluate $\int \dfrac{2\pi y}{2y^3-1}dy$

Evaluate: $$\int \dfrac{2\pi y}{2y^3-1}dy$$ I've been struggling with this for a while. If it had just been $y^3$ instead of $2y^3$ in the Denominator, Partial Fraction Decomposition, although ...
1
vote
0answers
52 views

Integration of certain real functions using Euler's Formula.

I've heard about using Euler's formula $$e^{ix}=\cos(x)+i\sin(x)$$ to transform rational functions of sine and cosine into computable indefinite integrals. However, upon attempting to apply this ...
3
votes
4answers
94 views

How to solve $\int e^{-\sqrt{x}}dx$

I have this integral: $$\int e^{-\sqrt{x}}dx.$$ This is what I have done: $$\int e^{-\sqrt{x}}dx = \int \frac{1}{e^{\sqrt{x}}} dx$$ I Tried to solve it by substitution: $$t = \sqrt{x}$$ $$ t^2 = ...
1
vote
1answer
25 views

Substitution needed for calculating an integral

Can you find two functions: $\phi:(0,\infty) \longrightarrow (0,\infty) $, $f:(0,\infty)\longrightarrow \mathbb{R}$, with $\phi$ differentiable, such that $f(\phi(x))\phi'(x)=\frac{1}{x ...
0
votes
1answer
39 views

Integral of $f(x,\lambda)=\frac{x}{\sqrt{x-\lambda x^2}}$

How can I compute the indefinite integral: \begin{equation} I=\int \frac{x}{\sqrt{x-\lambda x^2}}dx \end{equation} with $\lambda$ a positive real parameter? Thank you!
1
vote
0answers
92 views

A closed form for $\int x^nf(x)\mathrm{d}x$

When trying to find a closed form for the expression $$\int x^nf(x)\mathrm{d}x$$ in terms of integrals of $f(x)$ I found that $$\int xf(x)\mathrm{d}x=x\int f(x)\mathrm{d}x-\iint ...
4
votes
3answers
56 views

Solve it by simple method

Solve it by simple method $$\int \frac{\mathrm{d}x}{(x^2+1)^4}$$ This is what i did: Let $x=\tan{\alpha}$ After solving we get $\int \cos^6(\alpha)\;\mathrm{d}\alpha$ Again by expanding we ...
3
votes
3answers
81 views

What to do next with this indefinite integral?

I'd like to evaluate the following indefinite integral: $$\displaystyle\int \frac{dx}{(3+x^2)(\sqrt{1+x})}$$ I started by letting $y^2=1+x$ and, after simplifying, got here: $$ ...
4
votes
4answers
117 views

How to evaluate $\int \frac { \sin x+\cos x }{ \sin^4 x+\cos^4x}\, dx$?

How can one find $$\int \frac { \sin x+\cos x }{ \sin^4 x+\cos^4x}\, dx?$$
2
votes
2answers
99 views

How to find $ \int \frac{x^8}{e^{2 x}+1} \, dx$?

I need help with this $$ \int \frac{x^8}{e^{2 x}+1} \, dx$$ Can't figure it out. I tried with partial integration, and I lose myself. If someone can help me. Thanks a lot.
0
votes
5answers
120 views

How do I solve $\int\sin^5 x$?

Should I transform $\int\sin^5 x$ into $\int (\sin^2 x)^2 \sin x \; \Bbb d x$ to solve it? Or should I use a different trigonometric identity?
2
votes
4answers
119 views

Integrate $\frac{1}{\sqrt{4-x^2}}$ [duplicate]

Evaluate $$\int\frac{1}{\sqrt{4-x^2}}dx$$ I had this question on my calc exam today, and I have no clue how it's done. I was trying to factor 4-x² to see if I could see any patterns but no luck. One ...
0
votes
2answers
55 views

How to solve this integral and have arccos(…) as a result?

$$\int {\sqrt{\csc^{2}x -1}} \, d(\cos^2x)$$ I need to solve this integral in order to arrive to a solution that looks like $x= \arccos(...)$ The main substitution is already done, I don't know how ...
1
vote
1answer
36 views

Reduction formula doubt.

If $$I_n = \int{(\frac{1}{a^2+x^2})^{n}}dx$$ Prove that:$$I_n = \frac{x}{2a^2(n-1)(a^2+x^2)^{(n-1)}}+\frac{2n-3}{2(n-1)a^2}I_{n-1}$$ I used Ibp but couldn't get such a relation. Please help me. ...
1
vote
2answers
50 views

Integration of $\int \frac {x\arctan x}{(\sqrt{1+x^2})^3}$

How can I evaluate $$\int \frac{x \arctan x}{(\sqrt{1+x^2})^3}\,dx$$ I think it must be done by parts, but I can't get any appropriate substitution or such.
1
vote
2answers
40 views

How to solve this indefinite integral with $\arctan$?

$$\int \frac{(x^2-1)\;\text{d}x}{(x^4+3x^2+1) \tan^{-1}{\left(\frac{x^2+1}{x}\right)}}$$ We should divide numerator and denominator by $ x^2 $ and put $z=x+\frac{1}{x}$ but I'm still not getting the ...
2
votes
4answers
73 views

Find error in integration of $\int \frac {\sin 2x}{\sin^4 x + \cos^4 x}$?

Find error in integration of $\int \frac {\sin 2x}{\sin^4 x + \cos^4 x}dx$? The answer is supposed to be ($\arctan \tan^2 x + C$), but I obtained ($-\arctan \cos2x + C$) as follows. Please identify ...
0
votes
4answers
42 views

Hint to solve $\int \frac{1-x^2}{x^2+x-2}$

I need to solve this: $$\int \frac{1-x^2}{x^2+x-2}$$ I tried to simplify the Denominator in this way: $$ x^2+x-2 = (x-1)(x+2)$$ so my new integral is: $$\int \frac{1-x^2}{(x-1)(x+2)}$$ So now I ...
0
votes
2answers
37 views

how i should solve this problem?

$\int \frac{\text{d}x}{(1+\sqrt{x})(x-x^2)}$ how i should solve this problem ? i think we should take $x=\sin^2(x)$ and then proceed but still not able to solve , please help !
0
votes
1answer
41 views

Solve $\int \frac{2x+1}{x^2-2x+2}$

I need to solve $$\int \frac{2x+1}{x^2-2x+2}dx$$ I have noticed that the Numerator is almost the derivate of the Denominator so I did this: $$\int \frac{2x-2+2+1}{x^2-2x+2}dx = \int ...