3
votes
1answer
52 views

Is this an acceptable way to integrate?

I am supposed to find: $$ \int \sec(1-x)\tan(1-x) dx $$ I then set $ u = \sec(1-x) $ $$ du = -\tan(1-x)\sec(1-x)\ dx $$ therefore $$ \frac{-du}{\sec(1-x)} = \tan(1-x)\ dx$$ Which when applied gives ...
11
votes
2answers
118 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot ...
0
votes
1answer
38 views

How can the integral of $|\sin(x)|$ be $-\cos(x)\text{sgn}(\sin(x))$?

Wolfram|Alpha tells me that $\int|\sin(x)| = -\cos(x)\text{sgn}(\sin(x))$ (which happens to also be its derivative), but I don't understand how this is possible, because the resulting function jumps ...
7
votes
4answers
124 views

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

I would like some guidance regarding the following integral: $$\int\frac{dx}{(x^2+1)\sqrt{x^2+2}}$$ EDIT: The upper problem was derived from the following integral ...
1
vote
3answers
44 views

Integrating $\;\int x^3\sqrt{x^2 + 2}\,dx$

Integrate the following: $$\int x^3\sqrt{x^2 + 2}\,dx$$ I understand how to do basic integration by parts but I don't know what to do with $\sqrt{x^2+2}$. Do I divide the $\sqrt{x^2+2}$ by 2 ...
1
vote
1answer
44 views

Integral of a function which yields a hyper-geometric function

Note that $n$ is an arbitrary constant. $$ \int(\sin^n(x))dx $$ I start by using the obvious integrating by parts and get: $$ \frac{d}{dx}[x\sin^n(x)] = \sin^n(x) + nx\sin^{n-1}(x)\cos(x) $$ $$ ...
0
votes
2answers
33 views

Volume of a horizontal cylinder using height of liquid

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank. We need to determine the amount of liquid ...
1
vote
1answer
19 views

Total Mass of a Spherical Object

Consider a spherical galaxy with volumetric mass density, at a distance $s$ from the center, is given by $$ \rho = \frac{k}{1+s^3} $$ where $k$ is a constant. Let $k = 25$. Determine the total mass ...
0
votes
3answers
65 views

Integral using height to find volume

How do you find the volume of a "pit" which is circular in horizontal cross-section, and parabolic in vertical cross-section using height by "sticking". "Sticking" is when we insert a dipstick through ...
1
vote
2answers
56 views

find $\displaystyle \int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$

find $\displaystyle\int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$ If I do Integration by parts, I end up with $\displaystyle\int e^{-2x-x^2}\hspace{1mm}dx$ Which I believe cannot be ...
0
votes
1answer
38 views

Solving an Integral - $ \int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt $

How do we solve $ \int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt \tag 1 $ to a finite form? $k,a,b,c$ are constants $at^2+bt+c$ does not guarantee equal roots always
1
vote
1answer
36 views

Evaluating an indefinite integral with an inverse trigonometric function

I'm really stumped on a homework problem asking me to evaluate $\int \frac{ln\ 6x\ sin^{-1}(ln6x)}{x}dx$, and after a few hours of trying different approaches I'd definitely be appreciative for a bump ...
4
votes
2answers
75 views

Evaluation of $\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $

How do we find $$\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $$ NB: It is not mandatory that $ax^2+bx+c$ has only a single root
2
votes
2answers
57 views

Integration techniques for $\int x^3\sin x^2\,dx$

I've learned a couple of methods of integrating, but I'm still not sure when to use which one. Example problem is \begin{align} \int x^3\sin x^2\,dx \end{align} I tried using a method where I set ...
5
votes
2answers
94 views

Evaluation of $\int \frac{x\sin(\sin x)}{x+5} \ dx$

How do we find $$\int \frac{x\sin(\sin x)}{x+5} \ dx\ ,$$ is there any way to take that $\sin x$ out from parent $\sin(\cdot)$ ?
0
votes
0answers
24 views

Integration by Parts on Square Matrix

Note : " ' " implies derivative w.r.t s Given Data in the Problem We have given matrix functions $R(s)_{3\times 3}$ and $K(s)_{3 \times 3}$. It has following relationships $R(s)^{'}_{3\times ...
3
votes
4answers
96 views

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$ $\bf{My\; Try}::$ Let $\displaystyle I = \displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$, Now let $\displaystyle \tan ...
0
votes
1answer
54 views

Indefinite integral $\int t \cdot \cos^3(t^2)dt$

I am having trouble integrating $$\int t \cdot \cos^3(t^2)dt$$ Progress I have made $u=t^2$ which makes the problem $1/2 \int \cos^3(u) du$. After writing that out I subsituted $v=\sin(u)$ ...
1
vote
0answers
32 views

Integration by Parts in matrices

Given Data in the question We have a given equation based on matrices as follows $\frac{\mathrm{d} R(s)_{3\times3}}{\mathrm{d} s}=R(s)_{3\times3}K(s)_{3\times3} \tag 1$ $\frac{\mathrm{d} ...
0
votes
4answers
48 views

Definite Integrations problems [closed]

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
195 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
66 views

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

How to calculate the following integral? Calculate $\displaystyle\int(1-\sin x)^2\cos x\,dx$.
3
votes
1answer
75 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
87 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 $\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$ [closed]

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
102 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 ...
5
votes
1answer
83 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 ...
2
votes
3answers
88 views

Calculus 2 Integral of$ \frac{1}{\sqrt{x+1} +\sqrt x}$

How would you find $$\int\frac{1}{\sqrt{x+1} + \sqrt x} dx$$ I used $u$-substitution and got this far: $u = \sqrt{x+1}$ which means $(u^2)-1 = x$ $du = 1/(2\sqrt{x-1}) dx = 1/2u dx$ which means ...
10
votes
1answer
120 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
81 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
71 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
37 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
42 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
60 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
46 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? ...
4
votes
0answers
57 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
73 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
93 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
33 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
2answers
61 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
89 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
116 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 \ $) ...
6
votes
2answers
109 views

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

I am having quite a difficult time integrating $$ \int \frac{\mathrm{d}x}{x^3-1} $$ My first approach was to apply a partial fraction decomposition $$ \int \frac{\mathrm{d}x}{x^3-1} = \int ...
7
votes
0answers
339 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 ...