6.) What is the sum of 3x² + x+8 and x?

Calculus 2. Please help

Anarel [89]

Answer:

$\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}}} \, dx = \infty$

General Formulas and Concepts:

Algebra I

Exponential Rule [Rewrite]: $\displaystyle b^{-m} = \frac{1}{b^m}$

Calculus

Limits

Right-Side Limit: $\displaystyle \lim_{x \to c^+} f(x)$

Limit Rule [Variable Direct Substitution]: $\displaystyle \lim_{x \to c} x = c$

Derivatives

Derivative Notation

Basic Power Rule:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Integrals

Definite Integrals

Integration Constant C

Integration Rule [Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

U-Substitution

U-Solve

Improper Integrals

Exponential Integral Function: $\displaystyle \int {\frac{e^x}{x}} \, dx = Ei(x) + C$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx$

Step 2: Integrate Pt. 1

[Integral] Rewrite [Exponential Rule - Rewrite]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \int\limits^1_0 {\frac{e^{-x^2}}{x} \, dx$
[Integral] Rewrite [Improper Integral]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \int\limits^1_a {\frac{e^{-x^2}}{x} \, dx$

Step 3: Integrate Pt. 2

Identify variables for u-substitution.

Set: $\displaystyle u = -x^2$
Differentiate [Basic Power Rule]: $\displaystyle \frac{du}{dx} = -2x$
[Derivative] Rewrite: $\displaystyle du = -2x \ dx$

Rewrite u-substitution to format u-solve.

Rewrite du: $\displaystyle dx = \frac{-1}{2x} \ dx$

Step 4: Integrate Pt. 3

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} -\int\limits^1_a {-\frac{e^{-x^2}}{x} \, dx$
[Integral] Substitute in variables: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} -\int\limits^1_a {\frac{e^{u}}{-2u} \, du$
[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}\int\limits^1_a {\frac{e^{u}}{u} \, du$
[Integral] Substitute [Exponential Integral Function]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(u)] \bigg| \limits^1_a$
Back-Substitute: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(-x^2)] \bigg| \limits^1_a$
Evaluate [Integration Rule - FTC 1]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(-1) - Ei(a)]$
Simplify: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{Ei(-1) - Ei(a)}{2}$
Evaluate limit [Limit Rule - Variable Direct Substitution]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \infty$

∴ $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx$ diverges.

Topic: Multivariable Calculus

7 0

3 years ago

Solve the following inequality l x - 5 l > 25

svetoff [14.1K]

Simplify both sides of the inequality.

x-5>25

 Add 5 to both sides.

x-5+5>25+5

x>30

8 0

3 years ago

Caroline's hair is

Ipatiy [6.2K]

The inequality can be used to determine how many months will it take for Caroline's hair to be at least as long as Trinity's hair is (1/5)x + 7 ≥ (1/10)x + 10

<h3>What is an equation? </h3>

An equation is an expression that shows the relationship between two or more numbers and variables. An independent variable is a variable that does not depend on other variables while a dependent variable is a variable that depends on other variables.

Let x represent the number of months.

For Caroline's hair:

Length = (1/5)x + 7

For Trinity's hair:

Length = (1/10)x + 10

For Caroline's hair to be at least as long as Trinity's hair, hence:

(1/5)x + 7 ≥ (1/10)x + 10

The inequality can be used to determine how many months will it take for Caroline's hair to be at least as long as Trinity's hair is (1/5)x + 7 ≥ (1/10)x + 10

Find out more on equation at: brainly.com/question/2972832

#SPJ1

8 0

2 years ago

X < 4 and x > -3: I need this in interval notation please.

maksim [4K]

x > -3 and x < 4

in interval notation

$x\in(-3,\ 4)$

>; < - a open circle on the number line and the parenthesis ( or )

≥; ≤ - a filled-in circle on the number line and the parenthesis [ or ]

8 0

3 years ago

One cylinder has a volume that is 8 Centimeters cubed less than StartFraction 7 over 8 EndFraction of the volume of a second cyl

mojhsa [17]

The correct equation is:

$216 = \frac{7}{8} \times x - 8$

The volume of second cylinder is 256 centimeter cubed

Solution:

Given that,

One cylinder has a volume that is 8cm less than 7/8 of the volume of a second cylinder

The first cylinder’s volume is 216 centimeter cubed

From given,

Let "x" be the volume of second cylinder

Therefore,

Volume of first cylinder = 7/8 of the volume of a second cylinder - 8

$216 = \frac{7}{8} \times x - 8$

$216 = \frac{7x}{8}-8\\\\216 = \frac{7x-64}{8}\\\\7x - 64 = 8 \times 216\\\\7x - 64 = 1728\\\\7x = 1728 + 64\\\\7x = 1792\\\\Divide\ both\ sides\ by\ 7\\\\x = 256$

Thus the volume of second cylinder is 256 centimeter cubed

6 0

3 years ago

Read 2 more answers

6.) What is the sum of 3x² + x+8 and x? - 9 ?