2
votes
3answers
80 views

Find $\int \dfrac{dt}{t-\sqrt{1-t^2}}$

Find $\int \dfrac{dt}{t-\sqrt{1-t^2}}$ MY APPROACH : Substitute $t = \sin x$ Multiply numerator and denominator by $\cos x+\sin x$ then rewrite everything in terms in $\sin2x$ and $\cos2x$, we ...
0
votes
1answer
30 views

how to prove the only difference between antidrivaties of a function is in their constants?

how to prove "If F is an antiderivative of f on an interval I , then the most general ...
-1
votes
3answers
73 views

Find $\int (\arcsin x)^2\hspace{1mm}dx$ [on hold]

Find $\int (\arcsin x)^2\hspace{1mm}dx$ $ $ How do we approach this problem
0
votes
1answer
30 views

Evaluating the indefinite integral $\int\frac{dx}{qx+c}$

Evaluate the indefinite integral (remember to use $\ln |u|$ where appropriate) $$\int\frac{dx}{qx+c}\qquad (q\neq 0) $$ I have no idea how to approach this. But here's what a have so far using ...
3
votes
2answers
94 views

Computing an indefinite integral

Let $\ P_n (x) = 1 + \frac{x}{1!} + \frac{x^2 }{2!} + \cdots + \frac{x^n }{n!} \ $ and $ I(x) = \int \frac{2n!\sin x + x^n }{e^x + \sin x + \cos x + P_n (x)}\, dx $ . (Where $\ n \to \infty \ $) ...
5
votes
2answers
88 views

How to Solve $ \int \frac{dx}{x^3-1} $

I am having quite a difficult time integrating $$ \int \frac{dx}{x^3-1} $$ My first approach was to apply a partial fraction decomposition $$ \int \frac{dx}{x^3-1} = \int \frac{dx}{(x-1)(x^2+x+1)} ...
0
votes
2answers
67 views

Find $\int \sin^{-1}\sqrt{x}\hspace{1mm}dx$ [on hold]

Find $\int \sin^{-1}\sqrt{x}\hspace{1mm}dx$ Can someone explain how to approach this problem
5
votes
4answers
89 views

Calculus Question: $\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx$

How to evaluate integral $$\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx?$$ I tried substitution $u^2=x^2-1$ and $u=\sqrt{x^2-1}+x$ but it turns out too complicated. Could anyone here help me to ...
1
vote
3answers
99 views

Find $ \int \frac{dx}{x\sqrt{1-x^4}}$

Find $\displaystyle \int \dfrac{dx}{x\sqrt{1-x^4}}$ I cannot figure out how start this problem, can anyone explain
0
votes
1answer
95 views

Integration with exponential

$$\int y\,e^{x^2}\,dy$$ I begin with $$\int e^{x^2}y\, dy$$ let $u=e^x$, $du=e^x\, dx$ how do I continue?
0
votes
0answers
51 views

Evaluate $\int\tan x\, dx$

$-1 = 0$ by integration by parts of $\tan(x)$ The solutions to the above problem state that you can't cancel the integrals on each side because they both have an unknown constant attached to them. ...
2
votes
1answer
61 views

$\int \frac{1}{y-1} dy = \ln |y - 1|$?

I read the following in a book on differential equations $$\int \frac{1}{y-1} dy = \log |y - 1|$$ If I put $\int \frac{1}{y-1} dy$ into Wolfram Alpha it gives $\log (y - 1)$, i.e. the argument of ...
2
votes
3answers
180 views

Simplifying expression and finding indefinite integral

(a) Simplify $$\Large \frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \quad.$$ (b) Hence find $$\Large \int\frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \mathrm{d}x$$ I tried to find a breakdown of the expression, but ...
1
vote
2answers
53 views

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal?

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal? I know both of these are the answer to $\int \csc x \space dx$, and I am able to work them out with calculus ...
1
vote
1answer
44 views

First and second derivatives of the function $f(x)=x\int_0^x e^{t^2}dt$

I haven't done calculus for a while so I need your help with these two exercises. I am not sure whether my solutions are correct so I'd really appreciate someone's feedback. $$ f(x)=x\int_0^x ...
2
votes
3answers
68 views

Struggling with $\int \frac{dy}{y\left(1 - \frac y2\right)}$

I know I need to use a partial fraction and suspect I will end up with 2 terms that end up as a natural log integral but I just can't work it out. $$\int \frac{dy}{y\left(1 - \frac y2\right)}$$ I ...
2
votes
2answers
59 views

Solve $\int (4x+2)\sqrt{x^2+x+1}\,dx$

Trying to solve this for a while now, so far I was able to come up without a proper answer. Problem : $\displaystyle \int (4x+2)\sqrt{x^2+x+1}\,dx$. I tried to take two common from $(4x+2)$ and ...
2
votes
3answers
58 views

How to Integrate $\int\frac{(x^2)}{\sqrt{7-x^2}}dx$.

I am trying to Integrate $\int\frac{(x^2)}{\sqrt{7-x^2}}dx$ and I have worked this problem a couple times and keep getting the same answer. So I will show my process and please point my errors out. ...
2
votes
1answer
46 views

Integrating $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I am trying to differentiate from my previous question, but I am having trouble in the finishing steps. I have the integral $\int\frac{\sqrt{16x^2-9}}{x}dx$. $$v=4x \hspace{15pt}dv=4dx$$ ...
1
vote
2answers
51 views

How to integrate $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I have the integer; $\int\frac{\sqrt{16x^2-9}}{x}dx$, and I am having trouble doing the trigonometric substitution. So for integrals in the from of $\sqrt{x^2-a^2}$ where $a$ is a constant is by ...
3
votes
2answers
89 views

Integrate $\int\frac{dx}{x\sqrt{x^2+1}}$

I would like to ask for some help regarding the following indefinite integral, tried integration by parts and trigonometric substitution which both brought me to ...
4
votes
6answers
157 views

Evaluate $\int\frac{\sqrt{1+x^2}}{x}\, dx$

Evaluate $$\int\frac{\sqrt{1+x^2}}{x}\, dx$$ I tried substitution but failed miserably.
0
votes
1answer
11 views

indefinite trig substitution integral

I am not able to understand how to get the solution for an integral. Substituting something like $\frac{1}{12}tan(\theta)$ seems to be the right thing to do, but I can't figure it out from there. ...
1
vote
3answers
88 views

Evaluate $\int \frac{6x+4}{x^2+4}dx$

Find$$\displaystyle \int \dfrac{6x+4}{x^2+4}dx$$ I'm not really sure where to begin with this one - I know the answer will probably involve an $\arctan$, but I am unsure on how to use $\arctan$ in ...
1
vote
5answers
92 views

Evaluate $\int \frac{e^{-2x}}{e^{-2x}-3}dx$

$$\displaystyle\int \dfrac{e^{-2x}}{e^{-2x}-3}dx$$ I'm not sure how to integrate this. What's the first step? I thought it was the common result where the numerator is the derivative of the ...
2
votes
2answers
89 views

Calculate an integral with $\sin$? [closed]

How can I calculate the following integral? $$ \int \left[ \sin((x+x_0)T)\times\sin((x-x_0)T) \right]^{2n} \, \mathrm{d}x $$ where $x_0$ and $T$ are constants. Please help, thank you for your help ...
3
votes
1answer
56 views

Trig Substitution Problem - Integration

Suppose I want to integrate this (I chose an easy one): $$\int \frac {dx}{\sqrt{x^2-4}}$$ Method 1: (Trig Substitution) $x=2\sec(\theta)$ $$\int \frac {dx}{\sqrt{x^2-4}}=\int ...
2
votes
0answers
24 views

Expressing indefinite integrals in terms of a predefined set of functions.

It is well known that some integrals of elementary functions cannot be expressed as elementary functions. I was wondering if it was possible to extend the set of elementary operators by some ...
4
votes
2answers
119 views

Trying to Integrate$ \iint xy\log|x-y|\, dy\,dx $

Hello I am trying to integrate $$ I:=\int_{a}^{b}\int_{a}^{b}xy\log\left(\,\left\vert\,x - y\,\right\vert\,\right) \,{\rm d}y\,{\rm d}x,\qquad 0 < a <b $$ for $x,y\in \mathbb{R}$. I added the ...
-1
votes
3answers
71 views

How to evaluate an integral of the form $\int \frac{dx}{-ax^2 + b}$?

I need to evaluate $\int \frac{dx}{-ax^2 + b}$ while both $a$ and $b$ are positive. I was blocked while I was trying $ x=\tan\theta $ which turned $ dx=\sec^2\theta\, d\theta $ This method didn't ...
3
votes
2answers
46 views

How to simplify the integral of $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$?

So I am trying to integrate this problem $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$, and my professor went over it in class and went from $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx \rightarrow ...
3
votes
1answer
66 views

How do I integrate $\int_{0}^{\frac{\pi^2}{4}}7\sin(\sqrt{x})dx$?

So, quick backstory. My semester just started and we are starting off by learning integration by parts. Which hasn't caused me much trouble except for this problem. ...
2
votes
1answer
46 views

solve this ordinary differential equation?

i have the differential equation $y'=\frac{y-x}{y-x+1}$, how i solve this? try: i tryed to substitute $u=y-x$, then $u=y-x\iff y=u+x\Rightarrow y'=u'+1$ then $y'=\frac{y-x}{y-x+1}$ become ...
0
votes
1answer
44 views

Partial fraction decomposition and polynomials?

This answer gives a really great explanation of why partial fraction decomposition works. However, the explanation implies that rational functions can be decomposed into a sum of fractions plus a ...
0
votes
2answers
50 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
5
votes
1answer
75 views

How to evaluate $\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$

How to evaluate: $$\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$$ I have done a substantial work on it: Let $x^5z^3=x^5-1$. So $$x^5(z^3-1)=1\implies ...
3
votes
4answers
72 views

Shorter way to integrate $\int \frac{x^9}{(x^2+4)^6} \, \mathrm{d}x$

$$ I=\int \frac{x^9}{(x^2+4)^6}\mathrm{d}x $$ Yeah I know, I can substitute: $$t=x^2+4\text{ or }2\tan\theta$$ So that: $$I=\frac12\int\frac{(t-4)^4}{t^6}\mathrm{d}t\text{ or } ...
7
votes
4answers
121 views

How to find $\int \frac{x\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$

$$I=\int x.\frac{\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$$ Try 1: Put $z= \ln(x+\sqrt{1+x^2})$, $\mathrm dz=1/\sqrt{1+x^2}\mathrm dx$ $$I=\int \underbrace{x}_{\mathbb u}\underbrace{z}_{\mathbb ...
0
votes
1answer
62 views

Integrals related to the function $F(x) = \int_1^x (e^t/t )\, dt$

I'm having some trouble with part of a problem from Apostol Volume 1(Section 6.26, Number 6). For completeness I'll include the whole question: A function $F$ is defined by the following indefinite ...
5
votes
4answers
103 views

Evaluate $\int{\sin^3(x)\cos^2(x)}dx$

I'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!
3
votes
0answers
52 views

Integration indefinite integral of multiple functions

I need help integrating $$\frac{x}{1-\exp(-x^2/a^2)}\exp((x-u)^2/2s^2)$$ wrt $x$, where $a$ and $u$ are constants
0
votes
2answers
46 views

Calculus long division $\int\frac{y^4+3y^2-1}{y^3+3y}\ dy$

I have a problem like this in my homework and want to see how to go by doing this problem. I understand the long division, but cannot get the partial fraction part. $$\int\frac{y^4+3y^2-1}{y^3+3y}\ ...
4
votes
4answers
74 views

Integration of some floor functions

Can anyone please answer the following questions ? 1) $\int$ $ \left \lfloor{x}\right \rfloor $ $dx$ 2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$ 3) $\int_0^2$ $\left ...
0
votes
1answer
98 views

Calculate the integral of $\sqrt{36\sin^2(2t)+6\cos^2(t)}$

During an arc length calculation I reached the following integral and I am having hard time calculating it: $$\int\sqrt{36\sin^2(2t)+6\cos^2(t)}\,dt=\sqrt{6}\int\cos t \sqrt{24\sin^2(t)+1}\;dt$$ ...
2
votes
3answers
102 views

About integrating $\sin^2 x$ by parts

This is about that old chestnut, $\newcommand{\d}{\mathrm{d}} \int \sin^2 x\,\d x$. OK, I know that ordinarily you're supposed to use the identity $\sin^2 x = (1 - \cos 2x)/2$ and integrating that ...
3
votes
3answers
448 views

How to find the antiderivative of $\frac{1}{x^2(1+x^2)}$?

How to find the antiderivative of $\dfrac{1}{x^2(1+x^2)}$? I recognized that this can be done with trigonometric substitution and I let $x = \tan(x)$ and ended up with $\dfrac{1}{\tan(x^2)}$; then I ...
0
votes
2answers
105 views

Evaluate $\int \frac{du}{(u^2+2)^2}$ [closed]

Someone can help me with some idea to solve the integrate $$\int \frac{du}{(u^2+2)^2}$$ I tried to solve it using trigonometric substitution, but it failed.
2
votes
2answers
135 views

How to evaluate the following indefinite integral? $\int e^{e^x}\mathrm dx$

I stumbled across this question: what's the value of the following integral? $$\int e^{e^x}\mathrm dx.$$ Furthermore I was required to demonstrate. On wolfram I got the result $\operatorname{Ei}(e^x)$ ...
1
vote
1answer
46 views

$\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty $ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
1
vote
1answer
77 views

Tough integral with many radicals

I am completed baffled with this integral $$\int\left[\dfrac{1}{x^{1/3}+x^{1/4}}+\dfrac{\ln(1+x^{1/6})}{x^{1/3}+x^{1/2}}\right]\mathrm dx$$ Any tips?