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Calculus and Analysis
Integral Calculus

What is going on with this integral $\int \frac{d x}{\sqrt{e^{2 x} - 9}}$

Leland Morrow 2022-06-19

What is the polar form of $(- 11, 121)$ ?

minwaardekn 2022-06-19

Converting between explicit, implicit and parametric function
1. Given an explicit function $y = \sin (x) + \cos (x)$ , how to convert it to the respective parametric functions $x = f_{1} (t)$ , $y = f_{2} (t)$ ?
2. Given parametric functions $x = \sin (t)$ and $y = \cos (t)$ , how to obtain the respective implicit function $f (x, y) = 0$ ?
3. Given parametric functions $x = 1 + 2 t$ and $y = 3 + 4 t$ , how to obtain the respective implicit function $f (x, y) = 0$ ?

watch5826c 2022-06-19

Show that $g$ is differentiable and find $g^{'} (x)$ , FTOC

tr2os8x 2022-06-18

How do you find parametric equations and symmetric equations for the line through the points (1, 3, 2) and (-4, 3, 0)?

migongoniwt 2022-06-18

Prove that the simple Simpson’s rule
$\int_{a}^{b} f (x) d x \approx \frac{(b - a)}{6} [f (a) + 4 f (\frac{a + b}{2}) + f (b)]$
is exact for all cubic polynomials.

Carolyn Beck 2022-06-17

Find the derivative of integral
$H (x) = \int_{3}^{\int_{1}^{x} \sin^{3} t d t} \frac{d t}{1 + t^{2} + \sin^{6} t}$

Brajesh Kumar2022-06-16

Evaluate double integral of $\int_{0}^{1} \int_{0}^{x} e^{\frac{x}{y}} d x d y$

juanberrio8a 2022-06-16

Heaviside and trig function integral $\int \sin (3 t) θ (t) d t$

lobht98 2022-06-16

What is the distance between the following polar coordinates?: $(1, \frac{π}{4}), (12, \frac{9 π}{8})$

seupeljewj 2022-06-16

How do you give parametric equations for the line through that is perpendicular to the plane $(1, 3, - 5)$ perpendicular to the plane $2 x - 3 y + 4 z = 11$ ?

manierato5h 2022-06-16

Euler's Method for series associates with a given series $\sum_{j = 0}^{\infty} (- 1)^{j} a_{j}$ the transformed series $\sum_{n = 0}^{\infty} \frac{Δ^{n} a_{0}}{2^{n + 1}}$ where $Δ^{0} a_{j} = a_{j}$ , $Δ^{n} a_{j} = Δ^{n - 1} a_{j} - Δ^{n - 1} a_{j + 1}$ , $j = 1, 2, \dots .$ .
I am given that $\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + - \dots = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 2^{2}} + \frac{1}{3 \cdot 2^{3}} + \frac{1}{4 \cdot 2^{4}} \dots$ .

I need to prove that $\sum_{n = 0}^{\infty} \frac{Δ^{n} a_{0}}{2^{n + 1}} = \frac{1}{2} + \frac{1}{2 \cdot 2^{2}} + \frac{1}{3 \cdot 2^{3}} + \frac{1}{4 \cdot 2^{4}} \dots$ , but I can not figure out how. I attempt to by induction and the base case is simple enough. However, when trying to use the induction hypothesis I get stuck.

If I assume that $\sum_{n = 0}^{k} \frac{Δ^{n} a_{0}}{2^{n + 1}} = \frac{1}{2} + \frac{1}{2 \cdot 2^{2}} + \frac{1}{3 \cdot 2^{3}} + \frac{1}{4 \cdot 2^{4}} \dots \frac{1}{(k + 1) \cdot 2^{k + 1}}$ and I add $\frac{1}{(k + 2) \cdot 2^{k + 2}}$ to both sides. Then I just need to show that $Δ^{k + 1} a_{0} = \frac{1}{k + 2}$ . However I don't see how I can.

So I know that $Δ^{k + 1} a_{0} = Δ^{k} a_{0} - Δ^{k} a_{1}$ . From the induction hypothesis I know that $Δ^{k} a_{0} = \frac{1}{k + 1}$ . But I would have to work with $Δ^{k} a_{1}$ recursively. So I am not sure how to prove this.

Ezekiel Yoder 2022-06-16

What is the polar form of (3, -27)?

Kiana Dodson 2022-06-15

Evaluate the integral
$\int_{V}^{} \underline{\nabla} \cdot \underline{r} d V$

polivijuye 2022-06-15

The cubic Bezier curve can be given in matrix form as
$P (u) = (u^{3} u^{2} u^{u} 1) (\begin{matrix} - 1 & 3 & - 3 & 1 \\ 3 & - 6 & 3 & 0 \\ - 3 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) (\begin{matrix} P_{0} \\ P_{1} \\ P_{3} \\ P_{4} \end{matrix})$
If a cubic Bezier curve is rotated by an angle 30 around x-axis what is the parametric function of the new Bezier curve?

Dale Tate 2022-06-15

Calculate:
$\int_{- 14}^{- 8} y d x$

Jayla Christensen 2022-06-15

What is the polar form of $(2, 3)$ ?

juanberrio8a 2022-06-15

Prove
$\int_{- a}^{a} f (x) d x = 0$
assuming $f (x)$ is odd.

Leland Morrow 2022-06-15

What is the distance between the following polar coordinates?:
$(3, \frac{17 π}{12}), (5, \frac{7 π}{8})$

Dania Mueller 2022-06-14

I am attempting to use Euler's method to express the following system of ODE's...
$\frac{d y_{i}}{d x} = c_{i} (y_{i - 1}^{3} - y_{i}^{3}) + f_{i} (for i = 1, \dots, n)$
using the discretised equations, which are presented below in matrix notation:
$y (t_{k + 1}) = C_{1} y (t_{k}) + C_{2} y^{3} (t_{k}) + f$
Note $f_{i} = 1$ when $i = 1$ , but otherwise equals 0 (this considers the initial condition). Whilst I am familiar with Euler's method, I am unsure of the steps to get to the first equation to the second. Can anyone assist me with this? Understanding this is necessary for a project I'm working on, so I would really appreciate any help.

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