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

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1answer
52 views

Solving integral $\int \arcsin x \cos x dx$

Can anyone give me a hint how to solve $\int \arcsin(x)\cos(x)dx $ ?
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2answers
33 views

Integral $\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$

Is this integral known to have a closed form? $$\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$$ Is there anything special about it?
-1
votes
1answer
33 views

Prove that $\int_0^{\pi} \sin^nx\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cos(n+2)xdx=0$

Prove that $$\int_0^{\pi} \sin^nx\cdot\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cdot\cos(n+2)xdx=0$$ with $n \in \mathbb{N}$ I think it's true, but I can't prove.
0
votes
1answer
36 views

Reduction formulae

$$I_{n}=\int x^n(1-x^2)^{\frac{1}{2}} dx$$ Show that $$(n+2)I_{n}=(n-1)I_{n-2}-x^{n-1}(1-x^2)^{\frac{3}{2}}$$ I can't seem to get this answer. Can someone please explain how to get to this? Thanks ...
0
votes
1answer
33 views

Trigonometric integral (arctg)

I have a problem with this integral. $$\int \text{arctan}(x-2)dx =\text{ }?$$ I tried integration by parts but it doesn't lead to right result.
1
vote
2answers
47 views

What is $\int \frac{1}{\sqrt{25y^2-10y-3}}dy$

$= \int \dfrac{1}{\sqrt{(5y-1)^2-4}}dy$ $=\int \dfrac{1}{\sqrt{u^2-4}}\dfrac{du}{5}, \quad U$ substitution $=\int \dfrac{1}{10\cos(\theta)} 2\cos(\theta) d\theta, \quad$ Trig substitution $= ...
1
vote
2answers
57 views

Find $\int\frac{dx}{2+\sqrt{x}}$ (using Integration by Substitution)

I used the substitution: $u=x$ $du=dx$ $2+\sqrt{u}=2+\sqrt{x}$ I then substituted the u into the equation: $\int\frac{1}{2+\sqrt{u}}du$ $=\int{(2+\sqrt{u})^{-1}du}$ I'm not too sure how to ...
2
votes
2answers
256 views

Trig substitution fails for evaluating $ \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx$?

Evaluate the integral \begin{equation} \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx \end{equation} Basically I could substitute: $t = \sin x$ and get: $$\int \frac{t}{t^2 +t + 1} dt$$ But, ...
0
votes
1answer
69 views

What is $\int \frac{e^x}{(e^x - 3)(e^{2x}+1)}dx$

I get a final answer of $\frac{3\ln \left | e^x -3 \right |}{10} - \frac{3\ln \left | e^{2x} +1 \right |}{20} +\frac{\ln \left | e^{2x} + 1 \right |}{20} + C$ but it seems off, any help would be ...
0
votes
5answers
52 views

What is the integral of $\frac{\sqrt{x^2 +4}}{x}dx$

I use trig substitution then get to this step but then I get stuck: $\int \frac{2\sec ^3\theta}{\tan \theta}d\theta$ anything I do seems to further complicate it. Thanks in advance.
2
votes
1answer
47 views

Equivalent of $\int_0^{\pi/2}\cos^n(\sin(x))dx$

Let $\displaystyle u_n=\int_0^{\pi/2}\cos^n(\sin(x))dx$. How can I find an equivalent of $u_n$ when $n\to\infty$ ?
1
vote
3answers
20 views

Indefinite integrals with rati0nal and polynomial functions and Substituion

I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would ...
0
votes
0answers
35 views

indefinite integral problem: help needed

What will be the integral with respect to $t$ of: $$\frac{dA}{dt} = cx(t)y(t),$$ where $c$ is a constant and $x$ and $y$ are functions of time ($t$). Is there any other method besides inegration by ...
2
votes
3answers
62 views

What is the integral of $\frac{\sqrt{x^2-49}}{x^3}$

I used trig substitution and got $\displaystyle \int \dfrac{7\tan \theta}{343\sec ^3\theta}d\theta$ Then simplified to sin and cos functions, using U substitution with a final answer of: ...
1
vote
2answers
31 views

What is the indefinite integral of $x^2\sqrt{1+x^2}$

I get this but I don't know if it is correct. I used a reduction formula for $\tan^{2n}(x)\sec^{3}(x)$. Any help would be appreciated. My Final Answer: $$\frac{\sqrt{x^2+1} x}{8}+\frac{\sqrt{x^2+1} ...
2
votes
1answer
26 views

Stuck on an integration question…

$$\int x^{-\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)dx$$ The answer I should get is $$2x^{\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)-4(x+4)^{\frac{1}{2}}$$ but I keep going wrong. Can someone show me how to ...
1
vote
2answers
58 views

How to solve $\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$?

$$\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$$ I tried to do it by substitution with no sucess. Anyone can solve it?
3
votes
2answers
44 views

Antiderivative of $\frac{\sqrt{4-x}}{x\sqrt{x}}$

I need help to find the antiderivative of the function $\displaystyle x \, \mapsto \, \frac{\sqrt{4-x}}{x\sqrt{x}}$ on $]0,4[$. I have tried the change of variables $u = \sqrt{4-x}$ but it didn't ...
0
votes
1answer
23 views

Can a Function have Multiple Valid Indefinite Integrals

Working with U-substitution, I have to integrate the following. $\int x\cos(x^2)\sin(x^2)dx$ From my understanding you can take the integral by substituting $u$ for either $\cos(x^2)$ or ...
2
votes
2answers
49 views

Integrating: $ \;\int \frac{1}{x^2+3x+2} dx $

How can I solve the following integral: $$ \int \frac{1}{x^2+3x+2} dx $$ Should I proceed by changing the variable (substitution)? or should I use integration by parts? Or another method ...
0
votes
2answers
36 views

Can someone help check if I evaluated my integral right?

The integral given to be evaluated is $\int 6x\cdot\mathrm{arctanh}(x)\,dx$. I tried to evaluated and got the following answer: $$3x^2\operatorname{arctanh}x + 3x ...
1
vote
1answer
28 views

Solving integral (by substitution?)

How do I solve the integral $\int \frac{1}{\sqrt{b-x^2}}$ where b is a constant ? I know that $\int \frac{1}{\sqrt{1-x^2}} = \arcsin(x)$ , so I guess I have to substitute somehow clever. Can you ...
1
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2answers
53 views

How do I evaluate $\int \cot^2x$? [duplicate]

I have an integral with $\frac{1}{\tan^2x}$ needed to be evaluated. But instead of searching online for the antiderivative of $\cot^2x$, how would i find it from first principles?
0
votes
3answers
31 views

How does integrating over absolute values work with definite integrals?

I have $ \int_0^\pi | \sin(x/2) | \, dx $, and according to Wolfram Alpha, the indefinite integral is: $$ -2\cos(x/2)\operatorname{sgn}(\sin(x/2)) + C $$ but the definite integral above evaluates to ...
1
vote
1answer
24 views

A level Integration question.

1a) Prove that $$e^x\operatorname{sech} x\equiv\frac{2e^{2x}}{e^{2x}+1}$$ b) Find $$\frac{d}{dx}[\arcsin(\tanh x)]$$ Simplify your answer as far as possible. c) Hence, or otherwise, solve $$\int ...
-1
votes
2answers
35 views

How to integrate $\int\frac{-x-1}{(x^2-2x+5)}dx$

How do you integrate $$\int\dfrac{-x-1}{(x^2-2x+5)}dx$$ ? I would be really grateful for an answer.
2
votes
1answer
45 views

How to apply the Chain rule when using standard integrals/differentials?

For example, $\frac{d}{dx} \,\arctan x = \frac{1}{1+x^2}$ and I know this because it is given to me in table of standard derivatives and integrals. But if I want to differentiate something like ...
1
vote
2answers
54 views

Algebraic Substitution Of Fractions

I already tried to putting the square root like this: $\sqrt{\frac{x}{5 + x}}$ but I dont know what to do next. $$\int \frac{\sqrt{x}}{\sqrt{5+x}}dx$$
0
votes
3answers
51 views

Trig Substitution Integral Question

My class is going over trig substitution, but I can't figure this one out, mostly because it's not in the correct form. Could someone help explain how to set up this problem? $$ \int \frac ...
1
vote
3answers
80 views

Evaluation of a general trigonometric integral

How can I evaluate the integral $$\int\sin^k(x)\ dx$$ in which I don't know if $k$ is an even or an odd number?
0
votes
5answers
69 views

Hints to prove the result of an integral [closed]

I need some hints to show that $$\int \frac{dx}{\alpha^2+x^2}=\frac{1}{\alpha}\arctan\left(\frac{x}{\alpha}\right)+C, $$ where $C$ is a constant. Thanks in advance!
1
vote
1answer
44 views

Explain why graph of f lies below the $x$-axis in interval $[4\pi/9,5\pi/9]$

$f(x)=(x+1)\sin(3x)$ Explain why the graph of f lies below the $x$-axis for values of $x$ in the interval $[4\pi/9, 5\pi/9]$ From what i know/understand I'd have to look at the function in two ...
0
votes
1answer
258 views

Help me complete finding the Reduction formula of $J_n=\tan^{2n} x \sec^3 x dx$?

Please don't mark my question as a duplicate of Find the reduction formula for the following integral. This question, was asked by a user and I was trying to answer this, I couldn't complete my work ...
1
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2answers
19 views

indefinite integration the result is -1 instead of $1 \over 2$

Below is from a book, When 0 $\le$ x < 1, F(x) = $\int_0^x$ t dt = $x^2 \over 2 $; When 1 $\le$ x < 2, F(x) = $\int_0^1$ t dt + $\int_1^x$(2-t) dt = -$x^2 \over 2$ + $2x$ ...
1
vote
4answers
65 views

$I=\int \frac{\cos^3(x)}{\sqrt{\sin^7(x)}}\,dx$

$$I=\int \frac{\cos^3(x)}{\sqrt{\sin^7(x)}}\,dx$$ I tried to write it as $$I=\int \sqrt{\frac{\cos^6(x)}{\sin^7(x)}}\,dx$$ And $$I=\int \sqrt{\frac{1}{\tan^6(x)\sin(x)}}\,dx$$ but it seems to go ...
0
votes
1answer
37 views

Integral $\int(\frac{1}{n+\cos x})^{\frac{3}{2}}dx$

Find an expression for this indefinite integral.I tried using some online calculators, but it is not coming. $$\int\left(\dfrac{1}{n+\cos x}\right)^{\frac{3}{2}}dx$$
1
vote
0answers
29 views

General solution to 3D linear 2nd order PDE using Wronskian and Integrals?

Using the Wronskian and Indefinite Integrals, I can write the solution to the general one dimensional second order linear non-homogeneous differential equation $$ y''+p(x)y'+q(x)y=g(x) \\ y^*(x)= ...
1
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0answers
39 views

Analytical Solution to an indefinite integral

I was reading about the Risch Algorithm on Wikipedia, and came across the example below, which was taken from Bronstein's "Symbolic Integration Tutorial". I do not currently have access to this ...
1
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3answers
93 views

Indefinite integration problem $\int {\frac{1}{1+ \tan^4 x}}dx$

The following problem came up in my last examination. $$ \int {\frac{1}{1+ \tan^4 x} dx}$$ The difficulty I was facing was that I wasn't able to find anything to substitute as there was nothing ...
3
votes
4answers
95 views

Evaluate $\int(x\sqrt{1-x^4})dx$

I've attempted this question with the substitutions $x=\sin(\theta)$ and $u = \sin^2(\theta)$ but then I got stuck. I think the main problem here is the power is too high. I'm not sure how to reduce ...
2
votes
2answers
76 views

How do I evaluate $\int u^m (1-u^2)^n du$?

What I've tried so far : $$\int u^m (1-u^2)^n du$$ $$u=\sin x \implies du= \cos x dx$$ $$\int \sin^{m}x \cos^{n+1}x dx$$ I have no clue on how to continue from here. Also, if the indefinite ...
2
votes
2answers
59 views

Take integral of $\int 1/x \,dz$ where $z=x+y$?

I want to integrate $$\int \frac{1}{x} \,dz$$ where the measure is the total differential of $z=x+y$ and $x,y\in\mathbb{R}$ are variables. I wonder if the result should simply be: $$\int ...
0
votes
1answer
31 views

Integration with substitution results in 1/du

Consider following integral: $$F(x) = \int \frac{1}{x}\cdot \frac{1}{ln(x)}dx$$ which we can substitute with $u = u(x) = ln(x)$: $$\int \frac{1}{x} \frac{1}{u}dx$$ then we can find $dx$: $$u'(x) ...
0
votes
0answers
82 views

For each of the following integrals find an appropriate trigonometric substitution of the form x=f(t) to simplify the integral.

$$ \int x\sqrt{7x^2+42x+59} dx $$ There were never any examples quite like this in class, so I'm clueless as to how to figure out which trig function to use.
1
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4answers
95 views

Evaluate $\frac{1}{4}\int \frac{dx}{(kx^{2}+ax+b)^{2}}$

I would like this integral: $$\frac{1}{4}\int \frac{dx}{(kx^{2}+ax+b)^{2}}$$ I used substitution $x+ \frac{a}{2k} = \frac{\sqrt{4kb-a^{2}}}{2k} \tan \theta$. This will reduce the denominator to $$ ...
0
votes
1answer
21 views

Indefinite integral fractional roots

I know the result of these indefinite integrals, but I don´t understand how the calculaton gets there: $$\int \frac{1}{\sqrt{x}}dx = 2 \sqrt{x}$$ $$\int \frac{1}{\sqrt[3]{x}}dx = ...
3
votes
3answers
52 views

Finding $ \int \sin^2(2x)/[1+\cos(2x)]dx$.

I am surprisingly having a bit of difficulty with an indefinite integral which is interesting since the integral I solved before is $$ \int \frac{1+\cos2x}{\sin^2(2x)} dx$$ The integral I am ...
1
vote
1answer
62 views

How to solve $\int \frac{1}{\sqrt{\frac{C}{x^2}-1}}\;dx\;\;$

How does one solve the following integral: $$\int \frac{1}{\sqrt{\frac{C}{x^2}-1}}\;dx\;\;,$$ where $C$ is some constant. Should substitution be used here?
0
votes
1answer
23 views

Find appropriate substitution for indefinite integral.

Find the indefinite integral, $$ I = \int \frac{8 - 2x}{\sqrt{6x - x^2}} dx\,. $$ I know this is a 'substitution' question, but I can't work out what to substitute. Please could you tell me the ...
-4
votes
2answers
39 views

Evaluating $\int\frac{\sqrt{1-x}}{x}\,dx$

$$\int\frac{\sqrt{1-x}}{x}\,dx$$ $$\int \:uv'=uv-\int \:u'v$$ $$u=\frac{\sqrt{1-x}}{x},\:\:u'=\frac{x-2}{2\sqrt{1-x}x^2},\:\:v'=1,\:\:v=x$$