Tagged Questions
9
votes
2answers
116 views
$-1 = 0$ by integration by parts of $\tan(x)$
I had a calculus final yesterday, and in a question we had to find a primitive of $\tan(x)$ in order to solve a differential equation.
A friend of mine forgot that such a primitive could easily be ...
2
votes
2answers
36 views
$\int\sin^2(Cx)\,dx$ from a manual - need proof
In the book of quantum mechanics I came across an integral which was supposed to be from a manual ($C$ is a constant):
\begin{align}
\int\limits_{0}^d \sin^2\left( C x \right)\, d x = ...
1
vote
2answers
65 views
Can you explain me this antiderivative?
Find the antiderivative of $\displaystyle \frac{e^{\frac{x}{2}}}{e^x+2e^{\frac{x}{2}}+5}$.
The book suggests a switch of variables. Let $t=e^{\frac{x}{2}}$. And so $x=2\ln(t)$.
The antiderivative ...
6
votes
3answers
105 views
How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$
$$\int \dfrac{1}{\cos x+\tan x }dx$$
This can be converted to
$$\int \dfrac{\cos x}{\sin x+\cos^2x}dx$$
But from here I get stuck. Using t substitution will get you into a mess. Are there ...
3
votes
3answers
101 views
$\int \frac{1}{\cos(x)}\,\mathrm dx$
could you help me on this integral ?
$$\int \frac{1}{\cos(x)}\,\mathrm dx$$
Here's what I've started :
$$\int \frac{1}{\cos(x)}\,\mathrm dx = \int \frac{\cos(x)}{\cos(x)^2}\,\mathrm dx = \int ...
1
vote
1answer
59 views
Differentiable functions without an antiderivative
Specifically, why is there no antiderivative, or any possible method of integrating (except numerically) say $\;e^{\csc(x)}$?
(I don't have my computer handy right now so I cant format the formula, ...
0
votes
1answer
64 views
How to solve integrals of type $ \int\frac{1}{(a+b\sin x)^4}dx$ and $\int\frac{1}{(a+b\cos x)^4}dx$
$$\displaystyle \int\frac{1}{(a+b\sin x)^4}dx,~~~~\text{and}~~~~\displaystyle \int\frac{1}{(a+b\cos x)^4}dx,$$
although i have tried using Trg. substution. but nothing get
1
vote
2answers
43 views
Integration of $\int(2-x/2)^2dx$
Got an exam tomorrow and my head is no longer working. Could someone walk through the integration of this function
$$\int\left(2-\frac x2\right)^2dx$$
I understand integration by parts and stuff ...
3
votes
1answer
54 views
integrating logarithm or x raised to a power?
$\int\frac{15}{x}dx$ would be 15$\int\frac{1}{x}dx$ = $15\ln|x|+c$.
This seems like a silly question but I'm feeling exceptionally dense today. Why would you apply the logarithm rule, why wouldn't ...
1
vote
4answers
48 views
Partial integration
We want to integrate
$$ f(x) = 2x \cos(x)$$
We use partial integration where $2x = g$ and $\cos(x) = f'$
I end up with
$$d (2x \sin(x)) - \sin(x) \cdot d2x$$
What confuses me is the term ...
3
votes
1answer
57 views
Is my interpretation of integration correct?
I thought of this when I was thinking about the $dx$ under the integral sign. So we have a function $y=f(x)$. Therefore, $dy/dx=f'(x)$, so $dy=f'(x)dx$. Now the graph of $f$ is split into small ...
4
votes
3answers
97 views
Integral of rational functions.
I want to evaluate this integral:
$$\int{\frac{ax+b}{(x^2+2px+q)^n}}dx$$
The book only says to integrate by parts $\int{\dfrac{1}{(x^2+2px+q)^{n-1}}dx}$,
for simplicity if $n = 2$ I get:
...
1
vote
2answers
67 views
Evaluate $\int \dfrac{1}{\sqrt{1-x}}\,dx$
Find $$\int \dfrac{1}{\sqrt{1-x}}\,dx$$
I did this and got $\dfrac23(1-x)^{\frac32} + c$
But a online calculator is telling me it should be $-2(1-x)^{\frac12}$
What one is on the money and if not ...
1
vote
1answer
52 views
Integration $\int \left(x-\frac{1}{2x} \right)^2\,dx $
$$\int\!\left(x-\frac{1}{2x} \right)^2\,dx $$
From U-substitution, I got $u=x-\frac{1}{2x},\quad \dfrac{du}{dx} =1+ \frac{1}{2x^2}$ , and $dx= 1+2x^2 du$
and in the end I come up with the answer to ...
6
votes
3answers
117 views
Integral of $\cot^2 x$?
How do you find $\int \cot^2 x \, dx$? Please keep this at a calc AB level. Thanks!
6
votes
4answers
92 views
Integrate ${\sec 4x}$
How do I go about doing this? I try doing it by parts, but it seems to work out wrong:
$\eqalign{
& \int {\sec 4xdx} \cr
& u = \sec 4x \cr
& {{du} \over {dx}} = 4\sec 4x\tan 4x ...
0
votes
2answers
52 views
An integral problem?
How do you integrate $e^{e^x}$? I was able to get it down to du/(ln u) but I wasn't able to go further. Thanks!
2
votes
3answers
45 views
Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& ...
3
votes
2answers
73 views
evaluate $\int\ln x\tan x\,dx$
How to evaluate $\int\ln x\tan x\,dx$ ?
I've tried to do integration by parts but after calculations it cancel out the main question.
6
votes
2answers
137 views
Need help solving - $ \int (\sin 101x) \cdot\sin^{99}x\,dx $
I have a complicated integral to solve.
I tried to split ($101 x$) and proceed but I am getting a pretty nasty answer while evaluating using parts.
are there any simpler methods to evaluate this ...
2
votes
1answer
58 views
Solving for $x$ in this simple differential equation?
$\dfrac{dx}{dt}=2\dfrac{\sqrt{2g(\sin c- \sin x)}}{\sqrt{l}}$. $g$, $c$, and $l$ are all constants. Is there a way to solve for $x$ in terms of $t$ here? Once I did separation of variables and plugged ...
7
votes
2answers
136 views
Integrate $2\int x^2\, \sec^2x \,\tan x\, dx$
$$
2\int x^2\, \sec^2x \,\tan x\, \mathrm{d}x
$$
How to solve this using integration by parts? WolframAlpha can solve it, but is unable to give a step-by-step solution, and has a different answer to ...
1
vote
1answer
31 views
Specifying if a function has an elementary integral
In Algorithms for Computer Algebra in the last chapter about Risch algorithm, the Rothstein-Trager method is applied to see if an elementary function has an elementary integral. For this, the ...
1
vote
0answers
94 views
Integrating $\int [n (T - x) ^{n - 1} - 1] dx$ for constants $T$ and $n$
It's been way too long, and I'm having trouble integrating a function (with a practical application) that should be easy to do with high school calculus. It seems very simple compared to the questions ...
8
votes
4answers
403 views
What is $\int\frac{dx}{\sin x}$?
I'm looking for the antiderivatives of $1/\sin x$. Is there even a closed form of the antiderivatives? Thanks in advance.
2
votes
2answers
106 views
What is the indefinite integral of $f(x) = \begin{cases} \sin x & x<\pi/4 \\ \cos x & x\ge \pi/4 \\ \end{cases}$
I'm trying to find the indefinite integral of
$$f(x) = \begin{cases} \sin x & x<\pi/4 \\ \cos x & x\ge \pi/4 \\ \end{cases}$$
In all of $\Bbb R$. It seems continuous at $\frac{\pi}4$ and ...
5
votes
1answer
129 views
Need a hint to evaluate the indefinite integral $\int\frac{e^x(2-x^2)}{(1-x)\sqrt{1-x^2}}dx$?
So, the question says I have to perform the indefinite integration
$$\int\frac{e^x(2-x^2)}{(1-x)\sqrt{1-x^2}}dx$$
I know that
$$\int e^x(f(x)+f'(x))dx=e^xf(x)+C$$
Since any other substitution (using ...
0
votes
0answers
56 views
3
votes
4answers
174 views
Integrals from MIT integration bee
$\int\frac{dx}{2+2\sin x+\cos x}$
$\int_0^{\infty}\frac{\ln x}{1+x^2}dx$
$\int\frac{dx}{x(1+x^3)}$
In general what is $\int \frac{dx}{a+b\sin x}$?
8
votes
1answer
216 views
Evaluate $\int\sin(\sin x)~dx$
I was skimming the virtual pages here and noticed a limit that made me wonder the following
question: is there any nice way to evaluate the indefinite integral below?
$$\int\sin(\sin x)~dx$$
Perhaps ...
1
vote
2answers
56 views
Simple question - Proof
How is $\frac{1}{2}ln(2x+2) = \frac{1}{2}ln(x+1) $ ?
As $\frac{1}{2}ln(2x+2)$ = $\frac{1}{2}ln(2(x+1))$, how does this become$ \frac{1}{2}ln(x+1)$?
Initial question was $ \int \frac{1}{2x+2} $
What ...
1
vote
2answers
95 views
Evaluate $\int\frac{1}{r \ln(r)} \ dr$
What is the antiderivative of
$$\int\frac{1}{r \ln(r)} \ dr$$
I'm trying to use substitution, but substituting $u=r$ doesn't help as that just changes the variable.
6
votes
4answers
164 views
Evaluating$\int {1\over1-\sin2x}dx$
$$
\int {1\over1-\sin 2x}dx = \int {1\over \sin^2 x-2\sin x\cos x+\cos^2x}dx = \int {1\over (\sin x-\cos x)^2}dx
$$
From here I get two different answers, depending on whether I factor out $\sin x$ ...
8
votes
0answers
191 views
Evaluating $\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$
How to integrate?
$$\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$$
I have no idea how to do it.
Tried to get some information from wiki, but its too hard :|
0
votes
1answer
74 views
Reduced formula: $\int x^m \cdot e^x dx$
Please, I need find a reduced formula to $$\int x^m \cdot e^x dx$$ by parts integration method, I've found that every time I compute the m is reduced so: $m-1$, $m-2$, $m-3$, but I couldn't found the ...
6
votes
3answers
217 views
Evaluating $\int \frac {\sqrt{\tan \theta}} {\sin 2\theta} \ d \theta$
I am trying to evaluate
$$\int \frac {\sqrt{\tan \theta}} {\sin 2\theta} \ d \theta$$
I tried rewriting it as $$\int {\sqrt{\tan \theta}} \cdot \csc(2\theta) \ d\theta$$
Supposedly letting $u = ...
1
vote
2answers
796 views
Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$
I am trying to prove $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$
This problem is a classic, but I seem to be missing one step or the understanding of ...
1
vote
1answer
214 views
Evaluating $\int e^{ax} x^b (1-x)^c \mathrm{dx}$
Edit: clarify question
The integrand looks kind of like a gamma density function, and kind of like a beta density function, so maybe it has a somewhat nice solution?
$$\int e^{ax} x^b (1-x)^c ...
3
votes
1answer
79 views
How do I integrate this expression: $\int{2x\,dx\over x^3+x^{2/3}}$?
How do I integrate this expression:
$\displaystyle\int{2x\,dx\over x^3+x^{2/3}}$?
31
votes
5answers
1k views
Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
I've got troubles in computing the below integral:
$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$$
I hope it can be expressed in elementary functions. I've tried simple substitution as $u=\sin(x)$ ...
1
vote
1answer
191 views
A problem on indefinite integral: $\int(\cos x)^m\sin(nx)dx$
If
$$I(m,n)=\int(\cos x)^m\sin(nx)dx,$$
how do I get $7I(4,3)-4I(3,2)$?
4
votes
3answers
135 views
Evaluating $\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$
How do I integrate this expression:
$$\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$$.I got this in a book.I do not know how to evaluate integrals of this type.
2
votes
0answers
81 views
$ \int \frac{f(x) \bar f'(x)- f'(x)\bar f(x) + g(x)\bar g'(x) - g'(x)\bar g(x) }{f^2(x) + g^2(x)} \ dx$ over $\mathbb{C}$
Evaluating
$$ \int \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2} \ dx$$
should just give $\frac{f(x)}{g(x)}$. Now I have a similar quotient over $\mathbb{C}$, at least it looks similar. It's of the form
...
3
votes
1answer
191 views
Evaluating $\int (x^6+x^3)\sqrt[3]{x^3+2}dx$
I am trying to evaluate:
$$\int (x^6+x^3)\sqrt[3]{x^3+2} \ \ dx$$
My solution:
$$\int (x^5+x^2)\sqrt[3]{x^6+2x^3} \ \ dx$$
Let $$(x^6+2x^3) = t^3 \ \ \text{and} \ \ (x^5+x^2) \ \ dx = ...
2
votes
2answers
72 views
Please help me integrate the following: $\int \frac{y^2 - x^2}{(x^2 + y^2)^2}dy$
I'm self-studying a Cramster solution and I came across this integral and I don't know what they've done with it. Help would be appreciated.
$$\int \frac{y^2 - x^2}{(x^2 + y^2)^2} ~dy.$$
0
votes
2answers
293 views
Please help me to evaluate $\int\frac{dx}{1+x^{2n}}$.
Find please
$$\int\dfrac{dx}{1+x^{2n}}$$
where, $n \in N$.
1
vote
3answers
118 views
Evaluating $\int \left(\frac{3}{5} - \frac{8}{x}\right) \ dx$
I am unsure how to get the antiderivative of
$$f(x) =
\frac3
5
−
\frac8
x$$
I know the answer is $-8\ln(|x|) + C + \frac{3x}{5} $ as it says in my textbook.
But I am unsure how I get something ...
3
votes
3answers
731 views
Integral of $\int{\frac{dx}{(\arcsin{x})\sqrt{1-x^2}}}$
I am having a problem solving an integral. I am stuck in an infinite loop. Integral is:
$$\int{\frac{dx}{\sqrt{1-x^2}\arcsin{x}}}$$
I have separated it in dv and u on this way:
$$u = ...
2
votes
6answers
277 views
Integration of $\int\frac{1}{x^{4}+1}dx$.
I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}dx$. Do I have to use trigonometric substitution?
1
vote
5answers
210 views
Need a fast way to evaluate these integrals
Undoubtedly, this question is so easy but I'd like to ask it. We know that the way in which the indefinite integrals like $\int P(x)e^{ax}dx$ and $\int P(x)\sin(bx)dx$ wherein $P(x)$ is an arbitrary ...
