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e^{i 2 \omega_{o} t}

Generated Graph

Solution by Steps

step 1

To integrate the function

6 \cos(1 + \sin(t))

from 0 to 3, we set up the definite integral

step 2

The integral is

\int_{0}^{3} 6 \cos(1 + \sin(t)) \, dt

step 3

We use numerical methods to evaluate this integral as it cannot be expressed in terms of elementary functions

step 4

The result of the numerical integration is approximately

-1.6335888917

Answer

-1.6335888917

Key Concept

Numerical Integration

Explanation

The integral of

6 \cos(1 + \sin(t))

from 0 to 3 does not have an elementary antiderivative, so we approximate it using numerical methods.

What is the value of

e^{i 2 \omega_{o} t}

?

Solution by Steps

step 1

To find the value of

e^{i 2 \omega_{o} t}

, we recognize it as an expression involving Euler's formula

step 2

Euler's formula states that

e^{ix} = \cos(x) + i \sin(x)

, where

x

is a real number

step 3

Applying Euler's formula to our expression, we get

e^{i 2 \omega_{o} t} = \cos(2 \omega_{o} t) + i \sin(2 \omega_{o} t)

Answer

e^{i 2 \omega_{o} t} = \cos(2 \omega_{o} t) + i \sin(2 \omega_{o} t)

Key Concept

Euler's Formula

Explanation

Euler's formula relates complex exponentials to trigonometric functions, which allows us to express

e^{i 2 \omega_{o} t}

in terms of cosine and sine functions.

e^{i 2 \omega_{o} t}

Generated Graph

Solution by Steps

step 1

To find the integral of

6 \cos(1 + \sin(t))

from 0 to 3, we set up the integral

step 2

The integral is

\int_{0}^{3} 6 \cos(1 + \sin(t)) \, dt

step 3

We evaluate the integral using a numerical method or a calculator, as the integral does not have an elementary antiderivative

step 4

The calculator provides the numerical result of the integral

Answer

-1.6335888917

Key Concept

Numerical Integration

Explanation

The integral of

6 \cos(1 + \sin(t))

from 0 to 3 does not have an elementary antiderivative, so we use numerical methods to approximate its value.

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