Tagged Questions

Question about finding the primitives of a given function, whether or not elementary.

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0
votes
4answers
41 views

Integral with radical in denominator: $\int \frac{dx}{x(x^2-1)^{3/2}}$

I tried trigonometric substitution but it got me nowhere, and I can't find any examples online which has a radical in the denominator and a factor of $x$ outside of it. Own attempt: $$\int ...
8
votes
3answers
195 views

Indefinite Integral of Reciprocal of Trigonometric Functions

How to evaluate following integral $$\int \frac{\mathrm dx}{\sin^4x+\cos^4x\:+\sin^2(x) \cos^2(x)}$$ Can you please also give me the steps of solving it?
0
votes
1answer
40 views

Integration $\displaystyle\int \frac{x}{x^2-5x+6}dx$

Evaluate the Integral: $$\int \frac{x}{x^2-5x+6}dx$$ I solved twice and once I got $$3\log\left|x-3\right|-2\log\left|x-2\right|+C$$ and I tried again and changed one step and I got ...
1
vote
2answers
38 views

Evaluating $\displaystyle\int\frac{du}{\sqrt{-xu^{2}+yu+z}}$

This integral is just a step in a much longer problem for physics, but I am having some trouble with it. $$\int\frac{ \mathrm du}{\sqrt{-xu^{2}+yu+z}}$$ $x$, $y$ , and $z$ are constants Also ...
4
votes
2answers
60 views

Prove reduction formula for $\int \cos^m (x)\sin^n (x) \, dx$

$$\displaystyle\int \:\sin^n\left(x\right)\cos^m\left(x\right)\mathrm dx=\frac{\sin^{n+1}x\cos^{m-1}x}{m+n}+\frac{m-1}{m+n}\int \:\sin^nx\cos^{m-2}x\,\mathrm dx$$ I have been trying to solve for ...
-2
votes
1answer
55 views

solve the integral$\int\frac{x}{(\csc x)-x-x^2}dx$

$\int\dfrac{x}{(\csc x)-x-x^2}dx$ Its first time I solve same this integration I'm not sure what I can do. I had though the let $u=x~dx$ and $dv=\dfrac{dx}{(\csc x)-x-x^2}$ I can't solved it.
6
votes
2answers
118 views

Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?

My professor says that writing this is convenient $$\int \frac 1x \mathrm{d}x = \ln|x| + C\tag{1}$$ but wrong, since it should be written as: $$\int \frac 1x \mathrm{d}x = \begin{cases}\ln x + C ...
4
votes
2answers
75 views

How solve $\int \frac{dx}{(x^2-x)^x}$ [on hold]

I want solve $$\int \frac{dx}{(x^2-x)^x}$$. thanks for help
1
vote
3answers
83 views

Evaluating $\displaystyle\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$

Evaluate the following limit: $$\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$$ Any advice on how to tackle this problem ?
2
votes
4answers
97 views

Evaluating $\displaystyle4\int \frac{\tan^2x\:\sec\:x}{\sec\:x\:+1}dx$

I was solving following integral $$\int \frac{\sqrt{x^2+4}}{\frac{x}{2}+1}dx$$ I think I need do a trigonometric substitution but I eventually end up with $$4\int ...
1
vote
1answer
56 views

Demostrate $\int \frac{dx}{(a\sin x+b\cos x)^{n}} = \frac{A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+c \int \frac{dx}{(a\sin x+b\cos x)^{n-2}}$ [on hold]

Demonstrate: $$\int \frac{dx}{(a\sin x+b\cos x)^{n}} = \frac{A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+c \int \frac{dx}{(a\sin x+b\cos x)^{n-2}}$$ $A,B$ are undetermined coefficients
2
votes
3answers
103 views

Where did I go wrong in this integration $\int\frac{\ln(1-e^x)}{e^{2x}}\,dx$

Here is the closest I've come to the answer Link to Wolfram equality not giving true as output NB! I marked where I'm unsure in $\color{red}{red}$ color. And please don't get startled because of the ...
1
vote
3answers
54 views

Evaluate $\displaystyle\int _{-1}^0\:\frac{\left(x+6\right)}{x^2+2x+2}\:dx$

Evaluate following integral: $$\int _{-1}^0\:\frac{\left(x+6\right)}{x^2+2x+2}\:dx$$ I tried to solve it but don't know how to do it, can anyone please help
5
votes
2answers
120 views

Evaluation of $-\int e^{\cos(t)}\sin(\sin(t)+t)\,dt $

How would I integrate this: $$-\int e^{\cos(t)}\sin(\sin(t)+t)\,dt $$ I have tried several methods but can't seem to work this out.
-2
votes
3answers
42 views

Evaluating $ \int\frac{x}{\sqrt{3-x^2-2kx}}\,dx $ [on hold]

I'm trying to evaluate this integral: $$ \int\frac{x}{\sqrt{3-x^2-2kx}}\,dx $$ where $k$ is a real parameter.
1
vote
5answers
95 views

Indefinite integral of $\frac{\sqrt{x}}{\sqrt{x}+1}$

For this I tried using the substitution technique, but it got me nowhere near the right answer. What my notepad looks like: $$f(x) = \dfrac{\sqrt{x}}{\sqrt{x}+1}$$ and $$F(x) = \int f(x) = ...
1
vote
0answers
61 views

Problem in understanding the process of finding antiderivative.

Antiderivative or indefinite integral is the family of functions the derivative of which gives the original function. Now, let's elaborate the process. Suppose $F(x)$ is the derivative of the ...
-1
votes
1answer
20 views

Evaluate the integral (using partial fractions maybe?) [duplicate]

Evaluate the following integral $\int{\frac{1}{(x+a)(x+b)}}$ (this might involve partial fraction decomposition, $\int{\frac{1}{x^2+x(a+b)+ab}}$ this is what my first step was)
-4
votes
2answers
101 views

Evaluating $\int 2x e^{x^2} \, dx$ [on hold]

Can I have some help or pointers on how I should evaluate following indefinite integral? $$\int 2x e^{x^2} \, dx$$
0
votes
1answer
30 views

Finding $f(x)$ from first and second derivitive

What is f(x) when $$f(1)=0$$ first derivative $$f(1)=1/2$$ second derivative $$f(x)=1/x^3$$ Currently i have tried where the second derivitive = first derivitive + constant at x=1 1/x^3 = f^i(x) + ...
5
votes
3answers
101 views

Integration of $1/(x^4 \sin x +2x)$

$$\int\frac{1}{x^4 \sin x +2x} dx\ $$ How to evaluate this integral. How to go about evaluating these integrals?
0
votes
0answers
18 views

The Area of Right Triangle and Integral

Consider that we are dealing with a right triangle with constant base ($B=B_1$ and $\frac{dB}{dt}=0$). The values (of $B=B_1$ and of $\frac{dB}{dt}=0$) are maintained by nature to be constant from ...
1
vote
5answers
119 views

What is indefinite integral actually - $\int f(x)dx$ or $\int_a^x f(t)dt$?

What is indefinite integral? This is the question that always perplexes me. First my book wrote that Indefinite integral of $f(x)$ is $F(x)$ if on differentiation, it gives $f(x)$. In fact it is ...
-1
votes
0answers
23 views

The Area of Right Triangles and Fundamental Theorem of Calculus

Consider that we are given a right triangle with constant base $B=B_1$ and $\frac{dB}{dt}=0$. The values (of $B=B_1$ and of $\frac{dB}{dt}=0$) are maintained by nature to be constant from the ...
3
votes
3answers
51 views

Exercise with $u$-substitution

How would I "see" this or go a head to solve it? I just can't understand it or see why I would go that way; how can I make myself able to see stuff like this? $$\int \frac{t}{ \sqrt{4-t^4}}dt $$ ...
-2
votes
1answer
44 views

Integrability of a Dirichlet Function

Let $f:[0,1]\to \mathbb{R}$ $$ f(x)=\begin{cases}x & x\ \mbox{is rational} \\ -x & x\ \mbox{is irrational} \end{cases} $$ Prove that the function $f$ is not integrable. Use darboux sums and ...
2
votes
2answers
29 views

Calculate the following intergral

I have to calculate the following integral $$ \int \sqrt[3]{1+x\ln{x}} * (1+\ln{x}) dx$$ I have thought about using the following notation: $$ t = {1+x\ln(x)} => x\ln{x} = t-1 $$ But here I ...
6
votes
2answers
89 views

Evaluating $ \displaystyle\int e^{x\sin x+\cos x}\left(\frac{x^4\cos^3 x-x\sin x+\cos x}{x^2\cos^2 x}\right)dx$

Evaluate $$\displaystyle \int e^{x\sin x+\cos x}\left(\frac{x^4\cos^3 x-x\sin x+\cos x}{x^2\cos^2 x}\right)dx$$ $\bf{My\; Try::}$ Let $$\begin{align}I &= \int e^{x\sin x+\cos ...
1
vote
1answer
38 views

Are there examples of nontrivial $f$ for which the antiderivative of $\tan\circ f$ is known?

I'm looking for an example of a function $f$ (apart from inverse trigonometric and linear functions) such that $\int\tan(f(x))dx$ is known. Special functions included in the typical CAS are acceptable ...
3
votes
2answers
46 views

Evaluating $\int (\sin x)^2\,dx$ [closed]

Evaluate $$\int (\sin x)^2\,dx$$ Anyone can guide me for this? Thank you!
1
vote
1answer
37 views

$\int \tan(x)\operatorname{tanh}(x), \operatorname{dx}$

I was wondering if there existed a closed form for $$\int \tan(x)\operatorname{tanh}(x), \operatorname{dx}$$ I don't think this integral has a closed form, but could it be evaluated over some points ...
0
votes
1answer
22 views

cubic integral roots

I am trying to find the integral roots (if they exist) of the following polynomial. Additionally, it would be helpful if someone could explain an algorithmic approach to solving this. $$ f(x) = 2x^3 ...
1
vote
3answers
62 views

Integral of $\frac{x}{\sqrt{1+x^5}}$

I am trying to calculate the following integral: $\displaystyle\int_0^\infty \frac{x}{\sqrt{1+x^5}}\, dx$ But I can't seem to find a primitive for that function. I was trying to find a good ...
3
votes
1answer
96 views

Evaluation of the integral $\int \sqrt{t^4-t^2 + 1}\,dt$

My friend took his Calculus $2/3$ test yesterday. One of the questions he had trouble with was this integral: $$\int \sqrt{t^4-t^2 + 1}dt$$ My attempt It seems rather clear that the only approach ...
1
vote
1answer
13 views

Simplifying an indefinite integral representing the calculation of an average

I have an equation for a term $z_i$: $$ z_i = \ln\frac{a_iR+p_i}{T_o* tan\theta_i } $$ This represents a value in a grid, at location $i$, with the grid representing a geographic area. To get the ...
0
votes
1answer
20 views

Integral of $ye^{-(x+1)y}$

Not sure where I'm going wrong on this one. $$\int{ye^{-(x+1)y}}\:dy$$ $$u = y \qquad du = dy$$ $$dv = e^{-(x+1)y} \qquad v = -\frac{e^{-(x+1)y}}{x + 1}$$ $$-\frac{ye^{-(x+1)y}}{x + 1} \times ...
0
votes
2answers
63 views

Explain each step to find $ \int \frac{\cos x\,d x}{1 + (\sin x)^{2}} $

I know the answer but I don't understand the steps to integrate. $$ \int \frac{\cos x\,d x}{1 + (\sin x)^{2}} $$
1
vote
3answers
50 views

Indefinite integral of $\frac{\arctan x}{x^2+1}$

EDIT: I was studying from a site that uses really ambiguous notation so I misread $\arctan\ (x)^2$ as $\arctan\ (x^2)$. Now I can see why the integral is actually $\frac{1}{2} \arctan^2\ x + c $. ...
1
vote
1answer
36 views

Finding the antiderivative of a real power of a rational function

my abilites in integration (applied) are very limited, so my question is: if i would like to find out how to approach a problem like finding the antiderivative of the function $f(x) = ...
3
votes
3answers
64 views

Evaluate $\int t^2 e^{-2i\pi nt}\,dt$

I need to get $$\int t^2 e^{-2i\pi nt}\,dt$$ I'm thinking to use integration by parts, but $\int e^{-2i\pi nt}\,dt$ is tripping me up. Can anybody help? Thanks!
4
votes
2answers
82 views

Evaluate $\int\frac{1}{1+x^6} \,dx$

I came across following problem Evaluate $$\int\frac{1}{1+x^6} \,dx$$ When I asked my teacher for hint he said first evaluate $$\int\frac{1}{1+x^4} \,dx$$ I've tried to factorize $1+x^6$ ...
1
vote
0answers
25 views

Integral of a matrix exponent

What is the analytic closed form expression of $\int e^{A_1+A_2s} \ ds \tag 1$ where A and B are constant skew symmetric matrices NB $A_1=\left( \begin{array}{ccc} 0 & -c_0 & b_0 ...
1
vote
2answers
48 views

Integration of $\int {{e^{ax}}\cos bx\cosh cx\,dx}$

This formula appears in Gradshteyn and Ryzhik as formula 2.674.4. However it's too simple to have been included in Victor Moll's set of proofs. I've attacked it using integration by parts using both ...
6
votes
3answers
49 views

Find $\int \ln(\tan(x))/(\sin(x) \cos(x))dx$

I was given this question in a review package, and it has me stumped: I started off using the identity $\tan(x) = \sin(x) / \cos(x)$ and then used the fact that $\sin(x) \cos(x) = .5\sin(2x)$ to ...
2
votes
5answers
78 views

Integration of $ \int \frac{2\sin x + \cos x}{\sin x + 2\cos x} dx $

How can I integrate this by changing variable? $$ \int \frac{2\sin x + \cos x}{\sin x + 2\cos x} dx $$ Thanks.
-2
votes
3answers
52 views

How can I integrate $ \int \frac{\sin x}{(\cos x)^2-4}dx $ [closed]

How can I integrate this by changing variable? $$ \int \frac{\sin x}{(\cos x)^2-4}dx $$ Thanks a lot for helping me!
0
votes
2answers
45 views

Integrate $ \int \frac{e^{2x} - 2e^x}{e^{2x} - 1} dx $ [closed]

How can I integrate: $$ \int \frac{e^{2x} - 2e^x}{e^{2x} - 1} dx $$ by changing variable? Thanks a lot!
10
votes
8answers
248 views

Avoiding the Integration Constant

I sometimes find writing and keeping track of the constants of integration a somewhat messy job. Yes, sometimes it's necessary but in many situations that I come across in my level of mathematics, it ...
5
votes
1answer
95 views

I am having trouble with this integral from the 2012 MIT Integration Bee

$$\int\frac{dx}{(1+\sqrt{x})\sqrt{x-x^2}} $$ Could someone explain to me how to integrate this integral. Thank you and cheers.
2
votes
1answer
80 views

how to integrate $\mathrm{arcsin}\left(x^{15}\right)$?

Integral by parts: $$ I = x\sin^{-1}\left(x^{15}\right) - \int\frac{15x^{15}}{\sqrt{1-x^{30}}}dx $$ then what? The answer by wolfram gives an answer contains hypergeometric ${}_2F_1$ function,because ...