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

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1
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0answers
76 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
85 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
66 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
85 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
52 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
148 views

Help in finding the integral

I need help finding the indefinite integral of $$\int\,\frac{x}{(7x - 10 - {x^2})^{3/2}}\,\text{d}x\,.$$
-4
votes
2answers
50 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
32 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
66 views

Calculation of integral using two different methods? [on hold]

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 ...
3
votes
3answers
193 views

Indefinite integration: $\int x^{x^2+1}(2\ln x+1)dx$

Find the value of the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$ My attempt: I tried by using integration by parts, but not working since $x^{x^2+1}$ keeps coming again and again. Then I tried putting ...
0
votes
2answers
60 views

Certain integration technique

What technique to follow when integration functions in the form: $$\sin ax\over \sin bx$$ $$\cos ax\over \cos bx$$ $$\sin ax \over \cos bx$$ I do believe that all these forms should have a similar ...
2
votes
1answer
47 views

Evaluating the indefinite integral $\int\sqrt{\cos2x}\sin^32x\,dx$

I have tried to integrate the following indefinite integral but I'm not sure if I get the right answer. Please tell me if I'm wrong and if so, please indicate what went wrong. $$ ...
1
vote
5answers
80 views

How to integrate ${x^3}/(x^2+1)^{3/2}$?

How to integrate $$\frac{x^3}{(x^2+1)^{3/2}}\ \text{?}$$ I tried substituting $x^2+1$ as t, but it's not working
0
votes
2answers
134 views

How to find $\int x^2e^{x^2}dx$?

How to find $\int x^2e^{x^2}dx$? I tried integration by parts following ILATE rule but it's not working.Please help!! What should I take as first function ? If it's not integrable can you atleast ...
9
votes
7answers
145 views

Evaluating the indefinite integral $\int\sqrt{16-9x^2}\,dx$

I need to solve the integral below, but I just can't figure how. $$\int \sqrt{16-9x^2}\,dx$$ I have tried to replace $9x^2$ with $16\sin^2\theta$. I get to a point where I have the function ...
3
votes
5answers
164 views

How to integrate $\int \frac{4}{x\sqrt{x^2-1}}dx$

In order to solve the following integral: $$\int \frac{4}{x\sqrt{x^2-1}}dx$$ I tried different things such as getting $u = x^2 + 1$, $u=x^2$ but it seems that it does not work. I also tried moving ...
1
vote
1answer
88 views

Antiderivative of $\frac{e^x}{\sqrt{1-x^2}}$

Can anyone help me find the following indefinite integral: $$\int{\frac{e^x}{\sqrt{1-x^2}} dx}$$ I cannot think of any transformation...
2
votes
4answers
79 views

Solve $\int\frac{8x+9}{(2x+1)^3}\,dx$.

Do I split $\displaystyle\int\frac{8x+9}{(2x+1)^3}\,dx$ into partial fractions? Or do I use $(2x+1)^{-3}$ by itself? Not sure what to do. Please advice. The answer given is ...
2
votes
5answers
102 views

Mistake in evaluating $\int\dfrac{dx}{\ln(x)}$

Evaluate: $$I=\int\dfrac{dx}{\ln(x)}$$ My attempt: $$$$ $$I=\int \dfrac{x'}{\ln(x)} dx$$Integrating by Parts,$$\dfrac{x}{\ln(x)}-\int\dfrac{x}{(\ln(x))'}dx$$$$=\dfrac{x}{\ln(x)}-\int ...
0
votes
0answers
72 views

$\int_0^b \ln(\sin(ax))dx$ [duplicate]

Problem: Evaluate $$\int_0^b \ln(\sin(ax))dx$$ Unfortunately I have no idea as to how to proceed with finding a closed form for the above Integral. The $a$ in the integrand made me think of ...
5
votes
5answers
97 views

$\int\dfrac{dx}{x^2(x^4+1)^{3/4}}$ [duplicate]

Evaluate $$\large{\int\dfrac{dx}{x^2(x^4+1)^{3/4}}}$$ I thought of rewriting this as $$\large{\int\dfrac{dx}{x^5(1+\frac{1}{x^4})^{3/4}}}$$ and substituting ...
1
vote
2answers
48 views

Confusion about the integral $\int dT/(1-T^2)$

From some reference on the internet we have the following real valued function and its derivative: $$ M(T) = \frac{\sqrt{1-T^2}}{1+T} \quad \Longrightarrow \quad \frac{dM}{dT} = - M/(1-T^2) $$ The ...
2
votes
0answers
39 views

Help Integrating $I=\int\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$

I am trying to integrate the following function involving the Normal CDF ($\Phi$). I actually need the definite integral $$\int^b_a\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$$ for $q+ra,q+rb >0$ but ...
0
votes
5answers
106 views

What is the mistake in doing integration by this method?

Integration Of a given function can be found out in many ways, For a specific function ∫1/xlogx, if we do integration by parts (∫f(x) g(x)= f(x) ∫ g(x)- ∫ [d/dx (f(x)) ∫g(x)] dx ) we get this way ...
1
vote
1answer
37 views

Fourier series of constant on $2\pi$ intervals

I want to find a fourier expansion of only sines representing $g(x) = 1$ on the interval $[0, \pi]$. So I extend the function on $[-\pi, \pi]$ such that it is odd, and calculate $$b_k = \frac 1\pi ...
2
votes
2answers
99 views

Integration of $\int \frac{(1 + x)\sin x}{(x^2 +2 x)\cos^2 x-(1 + x)\sin2x}dx$

The integral is $$\int \dfrac{(1 + x)\sin x}{(x^2 + 2x)\cos^2 x-(1 + x)\sin2x}dx.$$I've tried the problem by first multiplying both the numerator and denominator by $\sec^2 x$ but couldn't do justice. ...
-4
votes
3answers
63 views

Evaluate the following integral : $\int(x^2 + 3x + 5)dx$ [closed]

Evaluate the following: $$\int(x^2 + 3x + 5)dx$$ Don't even know where to start with this one. Any help would be appreciated.
3
votes
1answer
154 views

How to integrate this : $ \int \cot(5x) \tan(2x) \mathrm{d}x$

Methods to integrate this integral: $$\int \cot(5x) \tan(2x) \mathrm{d}x$$ I have tried several methods, step by step, and they have led to invalid results. Helpful hints or processes are welcome. ...
2
votes
1answer
82 views

What is the general solution for integrals of the form $\int\frac{\;\ln^{m}(x+n) }{(x+n)^{b}e^{\alpha (x+n)}} dx$?

I have this integral $$\int\frac{\;\ln^{m}(x+n) }{(x+n)^{a}e^{\alpha (x+n)}} dx;\;\;m,n\in\mathbb{N_{>0}};\;\;a\in\mathbb{Q};\;\;\alpha\in\mathbb{R_{>0}}$$ I've tried to solve it with ...
3
votes
3answers
241 views

Proper integration procedure

Is this the proper procedure for the below integral? $$\int xe^{(x^2)}dx$$ let $u=e^x$ $du=xdx$ $$=\int u^2du$$ $$={u^3\over 3}+c$$ $$={(e^x)^3\over 3}+c$$ Wolfram alpha gives me this answer ...
2
votes
3answers
121 views

Indefinite integral $\int{3x^2\over (x^3+2)^4}dx$

Question: How to solve this indefinite interal $$\int{3x^2\over (x^3+2)^4}dx$$ Attempt: $3\int x^2{1\over (x^3+2)^4}dx$ Let $u=x^2$ then $du={1\over (x^3+2)^4}dx$. Am I on the right track or ...
0
votes
0answers
55 views

Hamiltonian action: what does $d^3x$ mean?

A really quick one. What does the d^3x term mean in this first line of this link: http://physics.stackexchange.com/questions/64272/retrieving-maxwells-equations-from-the-minimum-action-principle Does ...
3
votes
1answer
57 views

Integration of Bessel function of the second kind

I need to calculate the below integration; $$\int Y_0(x) \, dx,$$ where $ Y_0(x) $ is a zero order Bessel function of second kind. This seems like a simple integration, however I could nowhere find a ...
2
votes
1answer
115 views

what is the integration of $\int \frac {dx}{dx}$

Today I saw a book in the bookstore that has the following integral on its cover: $$\int \frac {dx}{dx} = \frac {1}{d} \ln x + c$$ I don't understand the meaning of $\frac {1}{d}$. Also, $\frac ...
3
votes
4answers
282 views

Integration of $\frac{\sin x}{\sin 4x}$

Question: Solve the following integral: $$\int \frac{\sin x}{\sin4x}dx$$ Attempt: Using trigonometric identities to expand $\sin4x$, I obtained the integral: $$\int \frac{1}{4\cos x \cos2x}dx$$ ...
2
votes
2answers
55 views

Trigonometric integration 3.

$\int \frac{\sqrt{3cos2x-1}}{cosx}$ ATTEMPT: I did the following substitution Let $sinx=t$ $cosxdx=dt$ $I=\frac{\sqrt{3(1-2t^2)-1}}{1-t^2}=\frac{\sqrt{2-6t^2}}{1-t^2}$ Now let $t=\frac{1}{z}$ ...
1
vote
1answer
152 views

Finding the general form for $\int\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$

I've been trying to find the general form of the following integral $$ I(x;p,q,r,a,b)=\int\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx $$ where $a,p,q \in \mathbb{N}\setminus\{0\}$ and $b \in \mathbb{Q}$. ...
2
votes
4answers
106 views

Evaluate the Integral:$\int\frac{(1+e^x)^2}{e^x}\ dx$

Evaluate the indefinite integral $$\int\frac{(1+e^x)^2}{e^x}\ \mathrm{d}x$$ My attempt: Expand numerator: $$\int\frac{1+2e^x+e^{2x}}{e^x} \, \mathrm{d}x$$ divide $e^x$ by the numerator: ...
2
votes
4answers
68 views

Evaluate the Integral: $\int\ x^2\ e^{x^3}\ dx$

$\int\ x^2\ e^{x^3}\ dx$ Step one: eliminate the $x$'s from the problem. The only way this can be done is utilizing $u$-substitution $u=x^3$ $u=x^3$ $du=3x^2\ dx$ $3\int\ e^u\ du$ at this point ...
1
vote
0answers
62 views

Difficult integral $\int u\, \partial_x \left[(\partial_x u)^2 + (\partial_y u)^2\right]\, \mathrm{d}x$

I need to integrate $\int u\, \partial_x \left[(\partial_x u)^2 + (\partial_y u)^2\right]\, \mathrm{d}x$. In other words we need $w$ such that $\partial_x w = u\, \partial_x \left[(\partial_x u)^2 ...
3
votes
0answers
33 views

Can an integral be certifiably non-elementary?

If you want to convince somebody that a particular natural number is prime, you can hand them a primality certificate--a small bundle of data which can be used to efficiently generate a proof of ...
2
votes
2answers
75 views

How to find the integral $\int \frac{\sqrt{1+x^{2n}}\left(\log(1+x^{2n}) -2n \log x\right)}{x^{3n+1}}dx$?

How to evaluate the integral : $$\int \frac{\sqrt{1+x^{2n}} \, \left(\ln(1+x^{2n}) -2n \, \ln x \right) \, dx}{x^{3n+1}}$$ I have attempted an evaluation, but I am at a loss as to a useful result. ...
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votes
2answers
66 views

Find the value of the given integral

Find the value of the integral - $$\int \cfrac{\cos^3 x + \cos^5 x}{\sin^2 x + \sin^4 x}dx $$ EDIT : This is what I've tried $$\int\cfrac{\cos x (\cos^2 x + \cos^4 x)}{\sin^2 x + \sin^4 x} dx \\ ...
3
votes
2answers
106 views

How do I evaluate this integral :$\int \frac{1}{ \sqrt{1-x^2} } \arctan( \frac{\sqrt{1-x^2}}{2})dx$?

Is there somone who can show me how do I evaluate this integral :$$\int \frac{1}{ \sqrt{1-x^2} } \arctan\left(\frac{\sqrt{1-x^2}}{2}\right)dx$$ Note: I took : $x=\cos t$ , but it didn't work Thank ...
-2
votes
1answer
45 views

integration of $\frac{1}{\sqrt{\sin(x-a)\sin(x-b)}}$ [closed]

What is the solution? $$\int\frac{1}{\sqrt{\sin(x-a)\sin(x-b)}}dx$$ I have already tried to solve this integration. But I failed.
1
vote
2answers
27 views

Integrating the given function involving trigonometric functions

Find $\int \csc^{p/3}x \sec^{q/3} x dx $ Given - $(p,q \in I^{+} )$ and $(p+q=12)$ I tried to substitute $q = 12-p$ in the integral but didn't find anything satisfactory.
1
vote
2answers
75 views

How do I evaluate this integral :$\int \frac{\sqrt{-x^2-x+2}}{x^2}dx$?

Is there someone who can show me how to evaluate this integral: $$\int \frac{\sqrt{-x^2-x+2}}{x^2}dx.$$ I have tried many changes of variables but I haven't succeeded yet. Thank you for any help.
0
votes
2answers
75 views

Integrate along the vertical strip

I want to show that some integration with vertical line is bounded. function $f(\mu)$ is given by $$ f(\mu)=A^{-\sqrt{\mu}} \frac{(B_1-\sqrt\mu)}{(B_2-\sqrt\mu)(B_3+\sqrt\mu)} $$ where $f$ is defined ...
0
votes
2answers
71 views

Integrating $\int \frac{\sin^{-1}(x)}{\sqrt{1+x}}dx$ by parts

I have a question that requires me to integrate the following by parts. I have done the question but apparently my answer does not match that of wolfram alpha's. $$\int ...
0
votes
2answers
34 views

Solve linear differential equation

So I have the following linear differential equation $$t\frac{dy}{dt}-3y=t^4$$ My first step was to divide through by $t$ to give $$\frac{dy}{dt}-3t^{-1}y=t^3$$ Then to find the integrating factor ...