Determine: <msubsup> ∫ <mrow class="MJX-TeXAtom-ORD"> −<!--

Taniyah Estrada 2022-06-15 Answered

Determine:

\int_{- 1}^{1} \frac{1 - x}{1 + x} d x

Ask Expert 1 See Answers

You can still ask an expert for help

Answers (1)

Xzavier Shelton

Answered 2022-06-16 Author has 26 answers

Let

a \in (- 1, 1]

F (a) = \int_{a}^{1} \frac{1 - x}{1 + x} d x

= \int_{a}^{1} (\frac{2}{1 + x} - 1) d x

= [2 \ln (1 + x) - x]_{a}^{1}

= 2 \ln (2) - 1 - 2 \ln (1 + a) + a

lim_{a \to - 1^{+}} F (a) = + \infty

the integral is divergent.

This is helpful

This is helpful 0

You might be interested in

asked 2022-04-08

What is the equation of the line normal to

f (x) = \frac{1}{x - 2}

See answers (1)

asked 2022-06-26

Where is the absolute value when computing antiderivatives?
Here is a typical second-semester single-variable calculus question:

\int \frac{1}{\sqrt{1 - x^{2}}} d x

Students are probably taught to just memorize the result of this since the derivative of arcsin(x) is taught as a rule to memorize. However, if we were to actually try and find an antiderivative, we might let

x = \sin θ ⟹ d x = \cos θ d θ

so the integral may be rewritten as

\int \frac{\cos θ}{\sqrt{1 - \sin^{2} θ}} d θ = \int \frac{\cos θ}{\sqrt{\cos^{2} θ}} d θ

At this point, students then simplify the denominator to just

\cos θ

, which boils the integral down to

\int 1 d θ = θ + C = \arcsin x + C

which is the correct antiderivative. However, by definition,

\sqrt{x^{2}} = | x |

, implying that the integral above should really be simplified to

\int \frac{\cos θ}{| \cos θ |} d θ = \int \pm 1 d θ

depending on the interval for

θ

. At this point, it looks like the answer that we will eventually arrive at is different from what we know the correct answer to be.

See answers (2)

asked 2021-05-30

g (x) = \int_{0}^{x} f (t) d t

where f is the function whose graph is shown in the figure.
(a) Estimate g(0), g(2), g(4), g(6), and g(8).
(b) Find the largest open interval on which g is increasing. Find the largest open interval on which g is decreasing.
(c) Identify any extrema of g.
(d) Sketch a rough graph of g.

See answers (1)

asked 2022-03-17

What is the instantaneous velocity of an object with position at time t equal to

f (t) = ({(t - 2)}^{3}, \sqrt{5 t - 3})

See answers (1)

asked 2022-06-16

How do you find the instantaneous rate of change of the function

f (x) = x^{2} + 3 x - 11

See answers (1)

asked 2022-05-08

Antiderivative of

(x^{2} + c)^{- 3 / 2}

What method should be used to determine the antiderivative of this expression?
Edit: I have

c > 0

in the problem I'm working on.

See answers (1)

asked 2022-04-12

How do you use the limit definition of the derivative to find the derivative of

f (x) = 3 x^{2} + 3 x + 3

See answers (1)

New questions

A = - a_{x} + 6 a_{v} + 5 a_{z} a n d B = a_{x} + 2 a_{y} + 3 a_{x}

, find the vector projection of B on A

A = 5 a_{x} + 3 a_{y} + 2 a_{z}

B = - a_{x} + 4 a_{y} + 6 a_{z}

C = 8 a_{x} + 2 a_{y}

find the values of

α

β

α A + β B + C

is parallel to the y-axis.

lim_{x \to 0} ﻿ \frac{(x \csc 2 x)}{\cos 5 x}

Evaluate the integral:

\int 4 x \sec^{2} x d x

f

be continuous on

[a, b]

f (a) \neq f (b)

. Then for every

λ

f (a) < λ < f (b)

, there exists a

c \in (a, b)

f (c) = λ

.

Question:
Suppose that

f : [0, 1] \to [0, 2]

is continuous. Use the Intermediate Value Theorem to prove that their exists

c \in [0, 1]

f (c) = 2 c^{2}

Attempt:
I know that when we have the condition were

f : [a, b] \to [a, b]

, the method to prove that c exits, is the same method you would use to prove the fixed point theorem.

Unfortunately I don't have an example in my notes when we have

f : [a, b] \to [a, y]

. How would I use the IVT to answer the original question?

Simplify. Write the answer as a single term without negative exponents.

\frac{7 n^{-} 7 - 7 m^{-} 7}{m + n^{8}}

Find the cross product

a \times b

and verify that it is orthogonal to both a and b.
a = j + 7k
b = 2i - j +4k

v_{1}, . . ., v_{n}

be linealy independent vectors in

R^{m}

w = a_{1} v_{1} + . . . + a_{n} v_{n}

with real numbers

a_{1}, . . . a_{n}

, be linear combination of these vectors.
Prove if

a_{1} \neq 0

, then the vector

w, v_{2}, . . ., v_{n}

are linearly independent
So far, what I have thought is forming a formula like this:

x_{1} w + x_{2} v_{2} + . . . + x_{n} v_{n} = 0

v_{1}, . . ., v_{n}

is linearly independent vectors, so

x_{2} = . . . = x_{n} = 0

, then now we need to prove

x_{1} = 0

so these vector will be linearly independent. But from here, I don't know how to process further more in this problem, can anyone help me out with a way to solve this or a better alternative solution is appreciate.

How do I find a constant B>0, so that

‖ x ‖

\leq

B * {‖ x ‖}_{\infty}

x \in R^{n}

?
Not sure but I think I have to look at

x

\sum_{i = 1}^{n} x_{i} e_{i}

with the development of the x to the canonical unit vectors

e_{i}

. But how do I do that?

f (x) = \frac{20}{(x^{6} + x^{4} + x^{2} + 1)}

.
I need to show that for any

k \in [5, 20]

, there is a point

c \in [0, 1]

f (c) = k

.
Thanks in advance.