Questions on the evaluation of definite and indefinite integrals
1
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0answers
31 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 ...
1
vote
2answers
29 views
the integral of a function $\frac{1}{u(x)}$
What is the integral of this function :
$$\frac{1}{u(x)}$$
for exemple :
$$\int_a^b\frac{dx}{1-x}$$
3
votes
1answer
29 views
Integral question - $\int\frac{(x+6)\,dx}{4x-x^2}$
Integral question - $\int\frac{(x+6)\,dx}{4x-x^2}$
What I did is $$\int\frac{(x+6)\,dx}{x(4-x)}$$
then
$$\int\frac{(x+6)\,dx}{4x-x^2}= \int\left(\frac{A \,}{x}+\frac{B}{4-x}\right) dx$$
this is the ...
2
votes
2answers
32 views
Integral question - $\int\frac{(4-x)\,dx}{x^2+4x+8}$
Integral question - $$\int\frac{(4-x)\,dx}{x^2+4x+8}$$
To solve it I need to bring the numerator to be the derivative of the dominator right?
I need to do the trick that not change the integral any ...
0
votes
3answers
49 views
Integral question - $\int\frac{\sqrt{\tan(x)}}{{\cos^2(x)}}dx$
Integral question - $$\int\frac{\sqrt{\tan(x)}}{{\cos^2(x)}}dx$$
I see that $\frac{1}{\cos^2(x)}$ is the derivative of $\tan(x)$ so I set $t = \tan(x)$? or the whole square?
Thanks!
6
votes
2answers
53 views
Stieltjes Integral meaning.
Can anybody give a geometrical interpretation of the Stieltjes integral:
$$\int_a^bf(\xi)\,d\alpha(\xi)$$
How would we calculate?
$$\int_a^b \xi^3\,d\alpha(\xi)$$
for example.
1
vote
1answer
29 views
Integral question - $\int\frac{\sin\sqrt{x}}{\sqrt{x}}$
This is the integral : $$\int\frac{\sin\sqrt{x}}{\sqrt{x}}$$
I thinking about put universal identity but not sure, I know that $\sin(x) = \dfrac{2t}{1+t^2}$.
But what about the square root? Instead ...
0
votes
1answer
34 views
Substitution in an integral
I need to do this integral:
$\int{\left( -\frac{\partial f}{\partial \varepsilon } \right)}\,{{\varepsilon }^{3/2}}d\varepsilon$,
where
$f\left( \varepsilon \right)={{e}^{-\varepsilon ...
-5
votes
1answer
34 views
Help with this problem (Integration)
Let $f:[a,b]\rightarrow R$ Riemann integrable. Proof that:
\begin{equation}
\int_{a}^{b} f(x)dx=\lim_{n \rightarrow \infty} \frac{b-a}{n} \sum_{i=1}^{n} f(a+i\frac{b-a}{n})
\end{equation}
10
votes
2answers
63 views
Evaluating $\int_0^{\infty} \text{sinc}^m(x) dx$
How do I evaluate $$I_m = \displaystyle \int_0^{\infty} \text{sinc}^m(x) dx,$$ where $m \in \mathbb{Z}^+$?
For $m=1$ and $m=2$, we have the well-known result that this equals $\dfrac{\pi}2$. In ...
2
votes
4answers
80 views
Improper Integral:$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$
I want show that this improper integral convergence: $$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$$ please help me.
1
vote
2answers
21 views
Calculus Integral Rotation Circle
Is this equation the equation for a circle with radius r and shifted down R?
How would i find the bounds of the integral for this problem? or is the integral going to be in terms of r?
2
votes
1answer
20 views
Calculus Integral and Reduction formula
For the first part I was thinking that since the function is even and the integral is from -2 to 2 then the negative part will cancel out the positive part. Is that the right thinking?
for the ...
1
vote
2answers
24 views
integrals involving minimum function {a,1-a,b,1-b}
I could not compute this integral. please help out! $0\le a,b \le 1$
$$f(a) = \int_0^1 \min{(a,1-a,b,1-b)}db$$
1
vote
2answers
45 views
Triple integral problem involving a sphere
Let $R = \{(x,y,z)\in \textbf{R}^3 :x^2+y^2+z^2\le\pi^2\}$
How do I integrate this triple integral
$$\int\int\int_R \cos x\, dxdydz,$$ where $R$ is a sphere of radius $\pi$?
I have trouble ...
