Detailed Explanations for Calculus 2 Equations

How to solve: $\int \frac{e^{x}}{e^{2 x} + 3} d x$
I tried substituting $u = e^{x}$ , $d u = e^{x} d x$ , and so I end up with:
$\int \frac{1}{u^{2} + 3} d u$
This looks like the derivative $\arctan (x)$ , but I have a 3 instead of a 1. How can I solve this integral?

Devin Anderson 2022-06-07

What is the surface area of the solid created by revolving $f (x) = \frac{1}{x^{2}}$ for $x \in [1, 2]$ around the x-axis?

Gybrisysmemiau7 2022-06-07

How do I prove that:
$lim_{n \to \infty} \sqrt[n]{4^{n} + 9 n^{2}} = 4$

Leland Morrow 2022-06-07

What is the surface area produced by rotating $f (x) = \frac{\sin x}{\cos x}, x \in [0, \frac{π}{4}]$ around the x-axis?

Ezekiel Yoder 2022-06-06

Compute
$\sum_{k = 2}^{\infty} \frac{(- 1)^{k} \ln k}{k}$

boloman0z 2022-06-06

How do you find the differential dy of the function $y = \frac{x + 1}{2 x - 1}$ ?

Summer Bradford 2022-06-06

I have the series $\sum_{n = 1}^{\infty} \frac{1 - \cos (\frac{x}{\sqrt{n}})}{\sqrt{n}}$ and I need to show that it converges pointwise for every $x \in R$

vrotterigzl 2022-06-06

Prove that $(\sum_{j = 0}^{N} a_{j}) (\sum_{j = 0}^{N} b_{j}) = \sum_{j = 0}^{2 N} c_{j}$

Autumn Marie2022-06-06

H1isvqjq jqbzjqbhq2022-06-05

A golf ball is hit a horizontal distance of exactly 300 m. What is the maximum height the golf ball reaches in the air if it is launched at an angle of 25 degrees to the ground?

Finley Mckinney 2022-06-05

What is the arclength of $f (x) = \frac{1}{\sqrt{(x + 1) (2 x - 2)}}$ on $x \in [3, 4]$ ?

bulbareeh5kl 2022-06-04

How do you find the average value of the function for $f (x) = 1 + x^{2}, - 2 \leq x \leq 2$ ?

Kane Barr 2022-06-04

How do you use the disk method to find the volume of the solid formed by rotating the region bounded by y=2x and $y = x^{2}$ about the y-axis?

Felix Moore 2022-06-04

I want to calculate ${Res}_{2} z^{2} e^{\frac{1}{z - 2}}$

Cinnamon 2022-06-03

Evaluate the line integral ∫c(−𝑥 + 𝑦 + 𝑧) 𝑑𝑥 + (𝑥 + 𝑦 − 𝑧) 𝑑𝑦 + (𝑥 − 𝑦 + 𝑧) 𝑑𝑧 where C is the line segment from the point 𝑃(−1, 2, 3) to 𝑄(3, −2, 1).

kincirrboh7 2022-06-03

Let $f : [a, b] \to R$ be a continuous function and let $f (a) < f (b)$ . Then by intermediate value theorem
$(1) f ([a, b]) = [f (a), f (b)]$
$(2) f ([a, b]) \supseteq [f (a), f (b)]$
$(3) f ([a, b]) \subseteq [f (a), f (b)]$
$(4) f ([a, b]) \neq [f (a), f (b)]$
For example if I take the function $f (x) = x^{2}$ on $[- 1, 2]$ then $f ([- 1, 2]) = [0, 4]$ which is not equal to $[f (- 1), f (2)] = [1, 4]$ , also $[1, 4] \subseteq [0, 4]$ , so options 1 is incorrect. But how to use IVT here? Please help.

Mary Ashley 2022-06-03

Prove that
the following sequence is increasing
$e_{n} = {(1 + \frac{1}{n})}^{n}, n \geq 1;$

stud4kuj5bwn 2022-06-03

How do you find the average value of the function for $f (x) = \sqrt[3]{x}, 0 \leq x \leq 8$ ?

Kubjatkocn2cl 2022-06-03

How do you find the differential dy of the function $y = \sqrt{9 - x^{2}}$ ?

Monserrat Monroe 2022-06-03

Let $f$ be continuous on $[a, b]$ , where $a < b$ , such that $f (x) \neq 0$ for all $x \in [a, b]$ . Prove that there is a $c > 0$ , either $f (x) > c$ or $f (x) < - c$ for all $x \in [a, b]$ .

What I did,

Proving by contradiction, Let $f (x) > c$ for some $x_{1} \in [a, b]$ and $f (x) < - c$ for some $x_{2} \in [a, b]$ . By the Intermediate Value Theorem, there exists exists a number $z \in (a, b)$ such that $f (z) = 0$ , however this contradicts the fact that $f (x) \neq 0$ for all $x \in [a, b]$ . Hence $f (x) > c$ or $f (x) < - c$ for all $x \in [a, b]$ .

Hi, was wondering whether my proof is sound. Do help me verify! Thanks!

When you are dealing with any Calculus 2 homework, it is vital to have a look at the various questions and answers that will help you see whether you are correct in your approach to finding solutions. Even if you are dealing with analytical aspects of Calculus 2, it will be helpful as you are looking at provided equations and learn how the answers relate to original questions and problems specified.

Do not be afraid to take a look at the basic integration and related application if Calculus 2 does not sound clear or start with the Calculus 1 first.

Calculus 2

Calculus 2 Equations: Expert Solutions and More