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1 point) Find a function

f

and a number

a

such that \[ 2+\int_{a}^{x} \fra...

Feb 1, 2024

1 point) Find a function

f

and a number

a

such that

2+\int_{a}^{x} \frac{f(t)}{t^{2}} d t=6 x^{-3}

Solution by Steps

step 1

Given the integral equation

2+\int_{a}^{x} \frac{f(t)}{t^{2}} dt=6 x^{-3}

, we want to isolate the integral on one side

step 2

Subtract 2 from both sides to get

\int_{a}^{x} \frac{f(t)}{t^{2}} dt=6 x^{-3} - 2

step 3

From the asksia-ll calculation list, we have two possible expressions for the integral:

6/x^3 - 2

and

-(2 (x^3 - 3))/x^3

step 4

Both expressions are equal to the integral, so we can set them equal to each other:

6/x^3 - 2 = -(2 (x^3 - 3))/x^3

step 5

Simplify the right side to get

6/x^3 - 2 = -2 + 6/x^3

step 6

This confirms that the integral of

f(t)/t^2

from

a

x

is indeed

6/x^3 - 2

step 7

To find

f(t)

, we differentiate both sides of the equation with respect to

x

, using the Fundamental Theorem of Calculus, Part 1

step 8

Differentiating

6/x^3 - 2

with respect to

x

gives

f(x)/x^2 = -18/x^4

step 9

Multiply both sides by

x^2

to solve for

f(x)

f(x) = -18/x^2

step 10

Since the integral is from

a

x

, and we have no initial condition to find

a

, we can choose

a

to be any value that makes the integral valid. Let's choose

a = 1

for simplicity

Answer

f(x) = -18/x^2

and

a = 1

Key Concept

Isolating the integral and differentiating to find the function

Explanation

We isolated the integral from the given equation and then differentiated both sides with respect to

x

to find the function

f(x)

. We chose

a = 1

because the problem does not provide specific conditions to determine

a