Question about finding the primitives of a given function, whether or not elementary.
4
votes
3answers
90 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
61 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 ...
0
votes
0answers
20 views
Question About Indefinite Integrals
I`m trying to understand how should I evaluate this indefinite integral with this data on the integral :
the question is : "Draw the shapes on the plain blocked - by the data lines and evaluate"
:
1) ...
1
vote
1answer
92 views
How to solve this integral easily: $\int \frac{x\cdot \sqrt[3]{x+2}}{x+\sqrt[3]{x+2}} dx$
I am trying to solve this integral $$\int\frac{x\cdot \sqrt[3]{x+2}}{x+\sqrt[3]{x+2}} dx$$
I can do it by brute force (means using a substitution then long division and then substitutions again) but ...
6
votes
0answers
54 views
How can I calculate $\displaystyle \int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$
How can I calculate $\displaystyle \int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$
My Try:: $\displaystyle \int \frac{1}{3x+5}\left(\sec x\tan x \right)\,\mathrm dx$
Now Using Integration by Parts::
We ...
1
vote
1answer
49 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 ...
5
votes
3answers
86 views
Integral of $\cot^2 x$?
How do you find $\int \cot^2 x \, dx$? Please keep this at a calc AB level. Thanks!
5
votes
4answers
72 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
49 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
68 views
The indefinite integral $\int \frac{1+\cos(x)}{\sin^2(x)}\,\mathrm dx$
I`m trying to solve this integral and I did the following steps to solve it but don't know how to continue.
$$\int \frac{1+\cos(x)}{\sin^2(x)}\,\mathrm dx$$
$$\begin{align}\int \frac{\mathrm ...
6
votes
1answer
67 views
Finding the antiderivative of $\frac 1{(1-x^m)^n}$,with $n,m\in\Bbb N$
For $m,n \in \Bbb N$, find the antiderivative of $g:(0,1)\rightarrow\mathbb{R}$ defined by:
$$g(x)=\frac{1}{(1-x^m)^n}$$
Mathematica gives a result with functions we didn't learn about yet. The ...
2
votes
3answers
41 views
Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& ...
0
votes
2answers
42 views
Finding the indefinite integral $\int \frac{3x+2}{(6x^2+8x)^7}\,\mathrm dx$
I'm not too familiar with how to solve this. Could anyone present a step by step guide on how to get the answer?
$$\int \dfrac{3x+2}{(6x^2+8x)^7}\,\mathrm dx$$
1
vote
3answers
65 views
Solve the following integral using substitution only?
Can you solve the following integral using only substitution? $$\int \dfrac{dx}{\left(\sqrt{x^2-4}\right)^3}$$ I saw a solution to this which began with $x=2\sec(u)$, but is there another way to solve ...
5
votes
1answer
60 views
Absolute values in $\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$
in my math class we were given a list of indefinite integrals, and one of them was:
$$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$$
My working:
$$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}=\int ...
2
votes
2answers
63 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.
3
votes
1answer
44 views
How do I proceed with this integral?
$$ \int (2x)\ cos(5x)\ dx$$
I put
$u = 2x$
$du = 2\ dx$
$v = \frac{1}{5}sin(5x)$
$dv = cos(5x)\ dx$
Then I try $ uv - \int vdu $
$$ 2x \times \frac{1}{5}sin(5x) - \int 2\times\frac{1}{5}sin(5x)\ ...
5
votes
2answers
115 views
Need help solving - $ \int (\sin 101x) \cdot\sin^{99}x\,dx $
I have a complicated integral to solve.
I tried to split ($101 x$) and proceed but I am getting a pretty nasty answer while evaluating using parts.
are there any simpler methods to evaluate this ...
2
votes
1answer
56 views
Solving for $x$ in this simple differential equation?
$\dfrac{dx}{dt}=2\dfrac{\sqrt{2g(\sin c- \sin x)}}{\sqrt{l}}$. $g$, $c$, and $l$ are all constants. Is there a way to solve for $x$ in terms of $t$ here? Once I did separation of variables and plugged ...
8
votes
1answer
75 views
Integral Question - $\int\frac{1}{\sqrt{x^2-x}}\,\mathrm dx$
Integral Question - $\displaystyle\int\frac{1}{\sqrt{x^2-x}}\,\mathrm dx$.
$$\int\frac{1}{\sqrt{x(x-1)}}\,\mathrm dx =\int \left(\frac{A}{\sqrt x} + \frac{B}{\sqrt{x-1}}\right)\,\mathrm dx$$
This is ...
3
votes
2answers
46 views
Following integral?
How to solve the following:
$$\int_1^{x} \lfloor t\rfloor dt $$
I can conclude the answer is asymptotic to $\displaystyle \frac{1}{2} x^2 - \frac{1}{2} x$ and specifically it looks just like ...
2
votes
1answer
68 views
Evaluating $\int\frac{e^{-x^2}}{(1+2x^2)^2}dx$
I'm trying to evaluate the following indefinite integral:
$$
\int\frac{e^{-x^2}}{(1+2x^2)^2}dx
$$
According to Wolfram|Alpha, this integral evaluates to:
$$
\int \frac{e^{-x^2}}{(1+2 x^2)^2} dx = ...
3
votes
3answers
115 views
Evaluate the indefinite integral $\int\frac{dx}{(1+e^x)^2}$
Evaluate the indefinite integral $$\int\frac{dx}{(1+e^x)^2}$$
There is some clever trick to solve this, I think.
I'm really hesitant to ask a homework question without submitting an attempted ...
2
votes
1answer
41 views
A question about the derivative of $\arctan(f(x))$
We all know that:
$$ \int{\frac{f'(x)}{(f(x))^2 + 1}} dx = \arctan(f(x)) + c.$$
But what happens if we change the $1$ in the denominator? For example:
$$\int{\frac{f'(x)}{(f(x))^2 + c}} dx. \qquad ...
2
votes
2answers
68 views
Evaluating $\;\int\sqrt{2uv}~dv\;?$
How do you evaluate $\quad\displaystyle \int\sqrt{2uv}\,dv\;?$
I tried for half an hour using integration of parts and other "methods" but I can't seem to get the answer. I think it's the square ...
7
votes
2answers
122 views
Integrate $2\int x^2\, \sec^2x \,\tan x\, dx$
$$
2\int x^2\, \sec^2x \,\tan x\, \mathrm{d}x
$$
How to solve this using integration by parts? WolframAlpha can solve it, but is unable to give a step-by-step solution, and has a different answer to ...
4
votes
3answers
108 views
How to find $\int \frac{2x}{x^4+1}dx$
Can you give me a hint how to start solving this?
$$\int \frac{2x}{x^4+1} dx$$
1
vote
3answers
46 views
How to solve some simple integrals (HW)
I'm stuck with those integrals. Can you give me a hint how to start solving?
$$\int{\frac{\ln(x+1)}{x+1}}dx$$
$$\int{\frac{1}{x^2-1}}dx$$
3
votes
1answer
65 views
Computing the integral $\int \sqrt{\frac{x-1}{x+1}}\,\mathrm dx $
How do I compute the next integral:
$$\int \sqrt{\frac{x-1}{x+1}}\,\mathrm dx \;?$$
1
vote
1answer
28 views
Specifying if a function has an elementary integral
In Algorithms for Computer Algebra in the last chapter about Risch algorithm, the Rothstein-Trager method is applied to see if an elementary function has an elementary integral. For this, the ...
1
vote
0answers
92 views
Integrating $\int [n (T - x) ^{n - 1} - 1] dx$ for constants $T$ and $n$
It's been way too long, and I'm having trouble integrating a function (with a practical application) that should be easy to do with high school calculus. It seems very simple compared to the questions ...
4
votes
3answers
131 views
Finding the indefinite integral $\int\frac{\sqrt{x+8}}{\sqrt{x-3}-\sqrt{x+3}}dx$
This is a homework question, I tried many subtitutions but nothing worked for me...
$$\int\frac{\sqrt{x+8}}{\sqrt{x-3}-\sqrt{x+3}}dx$$
Any clue will help.
Thanks.
5
votes
2answers
159 views
The indefinite integral $\int x^2\sqrt{1-x}\,dx$
I'm trying get the integral $\int x^2\sqrt{1-x}\,dx$ but I don't know how to proceed. I know I have to use substitution, but that's it.
I tried to get some help with the wolfram alpha step-by-step ...
1
vote
4answers
55 views
how do we intergrate ln(x) I would like to know the steps, because I know the final answer, but confusing how to get it there
how do I take intergral of ln(x) I would like to know the steps, because I know the final answer, but confusing how to get it there
$\int$ ln(x) dx
is it intergration by parts ?
2
votes
2answers
76 views
Evaluate $\int \dfrac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}}\,dx$
Evaluate: $$ \space \space \int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}}\,dx $$
Please give proper directions/hints to evaluate this.
9
votes
4answers
399 views
What is $\int\frac{dx}{\sin x}$?
I'm looking for the antiderivatives of $1/\sin x$. Is there even a closed form of the antiderivatives? Thanks in advance.
3
votes
2answers
53 views
Indefinite integral of the inverse Pythagorean theorem?
So here is my equation:
$$\int{\frac{dx}{(x^2 + d^2)^{1/2}}}$$
Is there any way to solve this? Thanks! Also, $d$ is just a constant.
0
votes
2answers
45 views
Explanation of $\int\frac{r}{\sigma^2}\exp\big(\frac{-r^2}{2\sigma^2}\big)\; dr=-\exp\big(\frac{-r^2}{2\sigma^2}\big)$
Can you please explain this equality?
$$\int\frac{r}{\sigma^2}\exp\left(\frac{-r^2}{2\sigma^2}\right)\; dr=-\exp\left(\frac{-r^2}{2\sigma^2}\right)$$
Thanks a lot! :)
2
votes
2answers
104 views
What is the indefinite integral of $f(x) = \begin{cases} \sin x & x<\pi/4 \\ \cos x & x\ge \pi/4 \\ \end{cases}$
I'm trying to find the indefinite integral of
$$f(x) = \begin{cases} \sin x & x<\pi/4 \\ \cos x & x\ge \pi/4 \\ \end{cases}$$
In all of $\Bbb R$. It seems continuous at $\frac{\pi}4$ and ...
5
votes
1answer
123 views
Need a hint to evaluate the indefinite integral $\int\frac{e^x(2-x^2)}{(1-x)\sqrt{1-x^2}}dx$?
So, the question says I have to perform the indefinite integration
$$\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$$
Since any other substitution (using ...
0
votes
0answers
54 views
3
votes
4answers
162 views
Integrals from MIT integration bee
$\int\frac{dx}{2+2\sin x+\cos x}$
$\int_0^{\infty}\frac{\ln x}{1+x^2}dx$
$\int\frac{dx}{x(1+x^3)}$
In general what is $\int \frac{dx}{a+b\sin x}$?
3
votes
4answers
131 views
Integral of type $\displaystyle \int\frac{1}{\sqrt[4]{x^4+1}}dx$
How can I solve integral of types
(1) $\displaystyle \int\dfrac{1}{\sqrt[4]{x^4+1}}dx$
(2) $\displaystyle \int\dfrac{1}{\sqrt[4]{x^4-1}}dx$
-1
votes
0answers
39 views
Can you someone help me to find the indefinite integral, step by step. I did my self, and getting wrong answer. [duplicate]
can you please someone tell me how to do this indefinite integral in steps
$\int$$cos(\sqrt{6x})\over\sqrt{6x}$ dx
0
votes
2answers
76 views
what are the possible answers we can get for the below intergral?
Could you please tell me what are the possible answers (if there is more than one) for the following indefinite integral?
$$\int \dfrac{\cos(\sqrt{6x})}{\sqrt{6x}}dx$$
1
vote
0answers
56 views
Integration of Binomial Differentials Proof/Reference
In Piskunov's Calulus (P375 & P385) & Hardy's Integration of Functions of a Single Variable (P48) mention is made of Chebyshev's theorem on the integration of binomial differentials however no ...
2
votes
1answer
55 views
Integral of function
Find $\displaystyle\int {1\over s^2 (s-1)^2}\,ds$.
I'm not sure how to set the integral to something like $A/s^2+B/(s-1)^2$. I don't know when do we use $Ax$ and when do we use $Ax^2$ and when do ...
2
votes
1answer
56 views
Evaluating integral, first year calculus
We started integrals not too long ago, I understand it for the most part but I always have a problem figuring out how to solve ones involving trig identities. Like this:
$$\int ...
8
votes
1answer
205 views
Evaluate $\int\sin(\sin x)~dx$
I was skimming the virtual pages here and noticed a limit that made me wonder the following
question: is there any nice way to evaluate the indefinite integral below?
$$\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 ...
