Solve Integral Calculus Problems with Ease

College
Calculus and Analysis
Integral Calculus

Infinite amount of vertical asymptotes
Is it possible that the graph of function has infinitely many vertical asymptotes?
I suppose, that it is not possible, because such function would not exist. But I need to prove it in a math-fashioned-way, and I'm clueless how to do it.

Adrian Brown 2022-11-04

If $k = \frac{1}{1 + 2 x}$ , where x is an integer greater than 1 and k can be represented as a terminating decimal, find the sum of all possible values of k. Can anyone guide to me to find the solution for this problem?

Emmanuel Umar2022-11-03

Obtain the integral of the function $y = \int x c o s x d x$

kunguwaat81 2022-11-03

What are the vertical asymptotes for this function?
$F (x) = \frac{5 x^{2}}{4 x^{2}} + 9$
Okay so when i graphed this function there are no vertical asymptotes but why is that ? Becuase if u set the denominator equal to zero then u can solve and get the vertical asymptotes

Amy Bright 2022-11-03

I have an Euler method that has this form:
$\hat{I} (t_{n + 1}) = \hat{I} (t_{n}) + h β \hat{I} (t_{n}) [1 - \frac{\hat{I} (t_{n})}{N}]$
which can also be written like
$\hat{I} (t_{n + 1}) = ϕ (\hat{I} (t_{n}))$
where $ϕ (x)$ is the iteration function down below:
$ϕ (x) = x + h β x (1 - \frac{x}{N})$
I use $h = 6$ in this method but if I use a $h$ which is a little bit bigger (for example $h = 20$ ), I have an absolute instability error. I want to find the value of $h$ from which this absolute error is shown?

Laila Murphy 2022-11-02

Second Fundamental theorem of calculus
Theorem 3.20 (Second Foundamental Theorem of Calculus)
Let f be a continuous function on [a,b] and F any function on [a,b], differentiable on (a,b), continuous on [a,b] such that $F^{'} (x) = f (x)$ for all $x \in (a, b)$ .
Then
$\int_{a}^{b} f (x) d x = F (b) - F (a)$
I need to use the second Fundamental theorem of calculus to work out:
$\int_{0}^{\frac{π}{8}} \tan (2 x) d x$
Firstly it is clear that tan(2x) is continuous on $[0, \frac{π}{8}]$
Now $F (x) = - \frac{1}{2} \ln | \cos (2 x) | = - \frac{1}{2} \ln \cos (2 x)$
where $F^{'} (x) = f (x)$
To show that F(x) is differentiable $\forall x \in (0, 1)$ is it enough to say that as f(x) is continuous on (0,1) the derivative exists?

blackdivcp 2022-10-31

The title itself is self explanatory - I am trying to numerically solve an ODE with an initial value that has an infinite gradient. It seemed problematic to me and I am not certain as to how I should approach this.
e.g. $\frac{d y}{d x} = \frac{y}{\sqrt{x}}, y (0) = 1$
(Obviously this can be solved analytically but I would like to know if there is any numerical method that tackles problems like this)

podvelkaj8 2022-10-31

Integral check $\int \frac{6 x + 4}{x^{4} + 3 x^{2} + 5} dx$

ajanlr 2022-10-30

Evaluating $\int \frac{1}{x^{7} - x} d x$

Valery Cook 2022-10-29

About $\int_{0}^{π / 2} \arctan (1 - \sin^{2} (x) \cos^{2} (x)) d x = π (\frac{π}{4} - \arctan \sqrt{\frac{\sqrt{2} - 1}{2}})$

Jack Ingram 2022-10-29

Find the area of the surface generated by revolving the curve $x = \frac{y^{3}}{3}, 0 \Leftarrow y \Leftarrow 5$ , about the y axis. The area of the surface generated by revolving the curve $x = \frac{y^{3}}{3}, 0 \Leftarrow y \Leftarrow 5$ , about the y-axis is __ square units

Aryanna Blake 2022-10-29

Need help solving complicated integral $\int e^{- x} \cos 4 x \cos 2 x d x$

gasavasiv 2022-10-26

Calculating slant asymptotes of radical function
I'm trying to calculate the slant asymptotes of the function $\sqrt{x^{2} + 2 x + 2}$ . I've found out that the gradients of the asymptotes are 1 for $x \to + \infty$ and -1 for $x \to - \infty$ . I've also found out that the constant of the positive asymptote is 1. Intuitively, I know the constant of the negative asymptote is -1, but I'm struggling to show it through calculation. I need to evaluate this to find it:
$lim_{x \to - \infty} \sqrt{x^{2} + 2 x + 2} + x$
without using l'Hôpital's rule (for the purposes of the assignment I'm not supposed to know how to use it.) I have tried rationalizing the numerator using the conjugate but I just end up with an undefined value.

Antwan Perez 2022-10-25

Asymptotes and uniform continuity
Let $f : R \to R$ be a continuous function with oblique asymptotes for $x \to \pm \infty$ . Prove that the function is uniformly continuous.
Proof: Let . Since the function has obliquous asymptotes then
$\exists M > 0 : | f (x) - (a x + b) | < ϵ \forall x > M$
and
$\exists m > 0 : | f (x) - (c x + d) | < ϵ \forall x < - m$
Now, in the interval the function is uniformly continuous because of the Heine-Cantor theorem. We deal with the case $+ \infty$ since the other one is formally identical. We know that lines are uniformly continuous. So, using the that certainly exists for the line we have:
$\begin{aligned} | f (x) - f (x_{0}) | & = | f (x) - (a x_{0} + b) + (a x_{0} + b) - f (x_{0}) | \\ \leq | f (x) - (a x_{0} + b) | + | (a x_{0} + b) - f (x_{0}) | \\ < | f (x) - (a x_{0} + b) | + ϵ \end{aligned}$
But
$| f (x) - (a x_{0} + b) | \leq | f (x) - (a x + b) | + | (a x + b) - (a x_{0} + b) | < 2 ϵ$
So we are finished because if a continuous function is uniformly continuous in and , then it is uniformly continuous in .

Jairo Decker 2022-10-24

Can you please explain this equality?
$\int \frac{r}{σ^{2}} \exp (\frac{- r^{2}}{2 σ^{2}}) d r = - \exp (\frac{- r^{2}}{2 σ^{2}})$

Josiah Owens 2022-10-24

10.5 divided by 5.5 (I chose this problem because it will OBVIOUSLY have a remainder)
How should I convert the remainder to a decimal?

snaketao0g 2022-10-23

Use the residues theorem to calulate
$\int_{| z - i | = 3} \frac{e^{z^{2}} - 1}{z^{3} - i z^{2}} d z$

Christopher Saunders 2022-10-23

Compute the indefinite integral
$\int \frac{7 x^{13} + 5 x^{15}}{(x^{7} + x^{2} + 1)^{3}} d x$

Chaim Ferguson 2022-10-23

Let's say I have solved an ODE with Euler's forward method, and also solved it using RK4, in both cases for varying decreasing step sizes h. Is there any way to look at the graphs and "see" the order of accuracy of the methods?

ndevunidt 2022-10-23

I have this differential equation $y^{'} = \frac{y (\sin t)}{t}$ , $y (0) = 2$ , and $h = \frac{1}{4}$ . The first set of values, the inital, are (0,2). For the next iteration would it be $y_{1} = 2 + (\frac{1}{4}) (\frac{2 \cdot \sin (0)}{0})$ ? I know you can't divide by zero, so what would should I do in this case?

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