Find integral: \int e^{x}\sin e^{x}dx

Ayaana Buck 2021-08-15 Answered

Find integral.

\int e^{x} {\sin e}^{x} d x

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stuth1

Answered 2021-08-16 Author has 97 answers

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New questions

Convert the polar equation to a rectangular equation.
r = 2

Graphing functions with calculus
I am stuck on graphing

y = (x - 3) \sqrt{x}

I am pretty sure that the domain is all positive numbers including zero. The y intercept is 0, x is 0 and 3.
There are no asymptotes or symmetry.
Finding the interval of increase or decrease I take the derivative which will give me

\sqrt{x} + \frac{x - 3}{2 \sqrt{x}}

.
Finding zeroes for this I subtract

\sqrt{x}

and then multiply by the denominator

2 x = x - 3

.
This gives me 3 as a zero, but this isn't correct according to the book so I am stuck. I am not sure what is wrong.

Why should the gradient be 0 at a critical point?

How do you prove that the function

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A carousel has a radius of 17 ft and takes 28 seconds to make one complete revolution. What is the linear speed of the carousel at its outside edge?

For the transformation

L : C^{1} [a, b] \to

C [a, b]

L (f) = \frac{d f}{d x}

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f

x

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a) Show that

L

is linear.
b) Find

K e r (L)

n u l l i t y (L)

L

one-to-one? Explain.
e) is

L

invertible? Explain.

Finding asymptotes of a function using limits
I need to find asymptotes of a function. I didn't find any vertical asymptotes and I think there aren't any. But when trying to find a horizontal asymptote I evaluate the limit as x approaches infinity but can't successfully find it.

f (x) = x \arctan (x)

y^{'} (t) = f (t, y), a \leq t \leq b, y (a) = α

apply for the extrapolation technique
we let

h_{0} = \frac{h}{2}

and use Euler method with

w_{0} = α

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and then midpoint method

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to produce
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y_{1, 1} = \frac{1}{2} [w_{2} + w_{1} + h_{0} f (a + 2 h_{0}, w_{2})]

then why this form results

O (h_{0}^{2})

approximation to

y (t_{1})

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h

2 h

. The Richardson extrapolation gives an estimate of

e = max_{t} | y (t, 2 h) - y (t, h) |

of the error. When the error is smaller than the tolerance, we keep the result and start from

2 y (t, h) - y (t, h)

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h / 2

until we reach the tolerance.
(

y (t, 2 h)

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)
I don't understand why the Richardson extrapolation is mentioned. For what do I have to use it? Can I not just calculate

y (t, 2 h)

y (t, h)

and see the error?

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