0
votes
4answers
44 views

Definite Integrations problems [on hold]

If $f(x)= x^2 e^{x^2}$ then show that $f'(x)= 2xe^{x^2} + 2x^3 e^{x^2}$ and use this result to evaluate $$\int x^3 e^{x^2} \, dx$$ How can I use the result to evaluate the integral?
5
votes
2answers
157 views

Simplest way to integrate this trigonometric integral:

$$\int \frac{1}{1+\tan x}dx,$$ A substitution like $t = \tan x, \;dt = (1+t^2)dx$ etc. immediately comes to mind, but I find this method a bit lengthy with the partial fractions. Is there a more ...
0
votes
2answers
56 views

Calculate $\int(1-\sin x)^2\cos x\,dx$ [on hold]

How to calculate the following integral? Calculate $\displaystyle\int(1-\sin x)^2\cos x\,dx$.
3
votes
1answer
48 views

How to evaluate the following integral? $\int\frac1{1+\sqrt{\tan x}}\mathrm dx.$

Evaluate the following integral: $$\int\dfrac1{1+\sqrt{\tan x}}\mathrm dx.$$ I know this question has a solution, but I haven't the slightest idea how to do it.
4
votes
2answers
83 views

How to calculate indefinite integral involving infinite sums?

I want to calculate the following integral: $$ \int_{0}^{\infty}\left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots\right)\;\left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot ...
1
vote
0answers
43 views

Partial fraction help

I need Help figuring out how to solve the indefinite integral of $$\int{ -5x^3-2x^2+32\over x^4-4x^3 } dx $$ using partial fractions. Please help. Thank you! I have already checked the online ...
0
votes
2answers
63 views

Integrate $\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$ [on hold]

Evaluate$$\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$$ Solution through either contour integral method or indefinite integral method please!
3
votes
4answers
98 views

Integrating $\int \dfrac {dx}{\sqrt{4x^{2}+1}}$

$\int \dfrac {dx}{\sqrt{4x^{2}+1}}$ I've been up to this one for quite a while already, and have tried several ways to integrate it, using substituion, with trigonometric as well as hyperbolic ...
3
votes
0answers
56 views

Integrals $\int \frac{1}{\operatorname{arctanh}(x)} \, dx$ and $\int \frac{1}{\operatorname{arccoth}(x)} \, dx$

Do we know anything about this integrals? $$ \begin{align} I_1(x) = \int \frac{1}{\operatorname{artanh}(x)} \, dx \\ I_2(x) = \int \frac{1}{\operatorname{arcoth}(x)} \, dx \end{align}$$ Similar ...
2
votes
2answers
25 views

Multiple answer for integration of a function?

Q. $\int \left(\frac{sin2x}{sin^4x+cos^4x}\right)\:dx$ My method: $$\int \:\left(\frac{sin2x}{sin^4x+cos^4x}\right)\:dx=\int ...
0
votes
1answer
20 views

Using the shell method, find the volume of the solid by rotating the region bounded by the given curves

$$x=y^2+1$$ $$x=2$$ about y=-2 How would I set this up? This is what I have so far: $$V = \int_0^2 2 \pi (y+2)(y^2+1) dy$$ I am almost certain this is wrong. Especially with the limits of ...
9
votes
1answer
108 views

Integral of $\sqrt{x^3 + 8}$?

I have issues solving the following integral: $$\int\sqrt{x^3+8}~dx$$ I tried substitution and integration by parts, but with no use. I'm guessing I have to use some trigonometric substitution. ...
2
votes
1answer
80 views

compute the integral: $\int\frac{x^2-1}{x^4-4x^2-1} dx$

I am trying to compute the integral $$\int\frac{x^2-1}{x^4-4x^2-1} dx.$$ I tried to use partial fractions technique but I got $3$ difficult terms which I don't know how to compute them. ATTEMPT: ...
0
votes
2answers
44 views

Why $\int {dx \over x \sqrt{x+3}} \neq \int {2u\cdot du \over (u^2-3)u} \rvert_{u=\sqrt{x+3}} $?

When trying to integrate the following, I thought these where equal: $$\int {dx \over x \sqrt{x+3}} = \left.\int {2u\cdot du \over (u^2-3)u} \right|_{u=\sqrt{x+3}} $$ But they are not. If you set ...
0
votes
1answer
56 views

Integral with quadratic square root inside trigonometric functions

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ ...
1
vote
0answers
30 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
1
vote
3answers
38 views

finding an indefinite integral of a fraction

(a) Show that $\frac{4-3x}{(x+2)(x^2+1)}$ can be written in the form ${\frac{A}{x+2} + \frac{1-Bx}{x^2+1}}$ and find the constants $A$ and $B$. (b) Hence find ...
1
vote
2answers
58 views

Integral involving exponents

How do we integrate $\int e^{C_1\frac{u^2+1}{u^2-1}} \ du\tag 1$ I could not find a proper substitution to convert it to a normal available form so that I can get a closed form of integration. $C_1$ ...
0
votes
0answers
41 views

integration involving imaginary terms

How do we integrate forms of following type with imaginary terms involved? Can we get a closed form of it as result? ...
3
votes
0answers
49 views

Integration using exponent

What could be the techniques we need to use to solve this integration $\displaystyle \int\tan^2\theta\frac{\sin^2(\sec\theta\tan\theta)}{\sec^2\theta}d\theta \tag1$? How do I convert this in to a ...
0
votes
1answer
68 views

Integration with quadratic square root

What could be the techniques we need to use to solve this integration $\int\dfrac{s^2\sin^2\left(s\sqrt{ as^2+bs+c}\right)}{as^2+bs+c}ds$ ? Main issue here is the term inside $\sin^2()$. Very ...
0
votes
1answer
14 views

Proof regarding the primitives of periodic functions

Let $ f:R \to R $ be an integrable, periodic function. Prove that any primitive of such a function can be written as a sum of a periodic function and a function of the form $G(x)=ax$ where $a$ is a ...
3
votes
3answers
90 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
32 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 ...
0
votes
1answer
52 views

Integrals involving roots

I am bit stucked with an integration form while doing one of my proofs for a graphics application.Issue is I cant take out the terms from the trigonometric functions for a proper known integral ...
-1
votes
3answers
79 views

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

Find $\int (\arcsin x)^2\hspace{1mm}dx$ $ $ How do we approach this problem
0
votes
1answer
32 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
110 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
90 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)} ...
7
votes
0answers
329 views

Integration of product of functions(Special form)

Sir, I have been doing a proof related to one research topic. But after a long effort, I got ended up in a messy integration equation. Could you give me some suggestions to solve this equations? (Any ...
0
votes
2answers
68 views

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

Find $\int \sin^{-1}\sqrt{x}\hspace{1mm}dx$ Can someone explain how to approach this problem
5
votes
4answers
96 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
100 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
100 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
63 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
183 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
72 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 ...
4
votes
3answers
74 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
59 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
49 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
54 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
94 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
161 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
12 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
89 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
93 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
92 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 ...