Let [tex] f \ : \ \mathbb{R}\to \mathbb{R} [/tex] be a continuous function which satisfies ,

[tex]\implies \sf f(x) = \displaystyle \int_0^x f(t) dt . [/tex]

Then the value of [tex] f(ln \ 5)[/tex] is ? ​

=====================================================

Explanation:

According to the Fundamental Theorem of Calculus, we can say

[tex]\displaystyle f(x) = \int_{0}^{x} f(t)dt\\\\\\\displaystyle f'(x) = \frac{d}{dx}\int_{0}^{x} f(t)dt\\\\\\\displaystyle f'(x) = f(x)\\\\[/tex]

Then think about a function in which its derivative is itself. The first thing that should come to mind is the exponential function [tex]f(x) = e^x[/tex]. There aren't any other functions that fit the description. The proof of this is lengthy and involves ODE (ordinary differential equations), so I'm not sure how far along in your calculus class you are.

Anyways,

[tex]f(x) = e^x\\\\\\f(\ln(5)) = e^{\ln(5)}\\\\\\f(\ln(5)) = 5\\\\\\[/tex]

The exponential base 'e' cancels out with the natural log.

We are given with a function and a condition of the respective function too, consider the given what we have ;

[tex]{:\implies \quad \displaystyle \sf f(x)=\int_{0}^{x}f(t)\: dt}[/tex]

By using the variable changing property we can change the Integrand and it will yield ;

[tex]{:\implies \quad \displaystyle \sf f(x)=\int_{0}^{x}f(x)\: dx}[/tex]

Now , let assume that

[tex]{:\implies \quad \displaystyle \sf \int f(x)dx=F(x)}[/tex]

Now, differentiating both sides wr.t.x (using Newton - Leibnitz's rule for Differentiating a function under the integral sign)

[tex]{:\implies \quad \displaystyle \sf \dfrac{df}{dx}=\dfrac{d}{dx}\bigg\{F(x)\bigg|_{0}^{x}\bigg\}}[/tex]

[tex]{:\implies \quad \displaystyle \sf \dfrac{df}{dx}=\dfrac{d}{dx}\{F(x)\}-\dfrac{d}{dx}\{F(0)\}}[/tex]

[tex]{:\implies \quad \displaystyle \sf \dfrac{df}{dx}=\dfrac{d}{dx}\{F(x)\}}[/tex]

[tex]{:\implies \quad \displaystyle \sf \dfrac{df}{dx}=f(x)}[/tex]

So now, we have a function whose derivative is equal to the respective function, now if you differentiate both sides again you will get [tex]{\bf{f^{\prime \prime}(x)=f^{\prime}(x)}}[/tex] and similarly you can get [tex]{\bf{f^{n}(x)=f^{n-1}(x)}}[/tex] . And their is only one function which satisfies this condition for f(x) defined from [tex]{\mathbb{R}}[/tex] to [tex]{\mathbb{R}}[/tex] i.e the exponential function, you just can prove this fact easily by using the Maclaurin's series, so we now arrived on the conclusion that [tex]{\bf{f(x)=e^x}}[/tex] , so now [tex]{\bf{f\{ln(5)\}}}[/tex] will be [tex]{\bf{e^{ln(5)}}}[/tex] i.e hence on simplifying [tex]{\bf{f\{ln(5)\}=\boxed{\bf 5}}}[/tex]

This is the required answer

Note :- If you want to prove the fact that for [tex]{\bf f\: : \: \mathbb{R}\to \mathbb{R}}[/tex], such that [tex]{\bf{f^{n}(x)=f^{n-1}(x)}}[/tex] is [tex]{\bf{f(x)=e^x}}[/tex], you can prove it by using the Maclaurin's series, and if you want it's proof too, I will prefer you seeing once here ;

https://brainly.com/question/27031371

y=-4/3x
Give the slope and y-intercept

i need now help me i need help​

HELP PLEASEEEEE will choose brainiest answer

The mode,media,median in
6, 18, 10, 3, 9, 2, 6, 4, 5, 7?

2.
Which of the following is the common difference of the arithmetic
sequence 4, 7, 10, 13, ...?
0
b. 1
c. 2
d. 3
a.
What is the correct answer​

help? will give brainlist.

Janelle plants 14 carrot plants for every 6 tomato plants in her garden. What is the ratio of carrot plants to total plants in Janelle’s garden?

1/5 (x - 8) equals x - 16 ​

C(n)= -9/2(-4/3)^n-1 what is the second term in the sequence

What is an equation of the line that passes through the points (5,2) and (-5,-6) PLEASE ANSWER

Help they need to get done in 10 minutes

2.0 x 10-3
divided by 8.0 x 10

What are the average high and low temperatures (in degrees Celsius) for Nuuk in January?

what is 98x100+5-23+15

Solve using substitution.

4x + y = –11
4x + 4y = 16

DaMarion bought 7 Candy Bars at the Student
Store, plus he spent $2 on a drink. He spent a
total of $16. How much did he spend per candy
bar?

What is m∠G to the nearest tenth? A right angle triangle EFG. The length of EF is 28. 2 and GF is 45. 8. Angle GFE is given as 90 degrees

Convert the following equation into standard form. ​

Write/simplify an expression of the total value if jake has 20 coins

Identify the value of y.
9y+6 = 3y+3+7+26

OeowowowkwiHELP ASaPieisowowiiwiwiwiiwiwisnsnsnxxj

4. 30 +-30 = 0.
=
a. Write another sum of two numbers that equals 0.
b. Write a sum of three numbers that equals 0.
c. Write a sum of four numbers that equals 0, none of which are opposite
Please help!?

Create a math problem that involves number 5 fractions


PLA HELPPPP

how many melimeter in 15 kilometre​

Lucy bought a table on sale for $665. This price was 24% less than the original price.
What was the original price?

Ratios

Mrs. Gildner took a survey of the 30 students in her math class. Of the 30 students surveyed, 18 of them said that chocolate is their favorite ice cream flavor. 4 students listed vanilla as their favorite flavor. The remaining students indicated that strawberry is their favorite kind of ice cream.


What is the ratio of chocolate to strawberry? SIMPLIFY if possible!

Which of the following results in the expression 0.22 x 86?
22 of 86%
22% of 86
86% of 0.22
0.86 of 22%

oH saVe mE sisTerS

Solve the inequality
29-2(3-5w) 13

HELP ASAP!!!!!! 65 POINTS!!!

PLEASE HELPPP!!! ASAPPP
WILL GIVE 20 POINTSSSSS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Question 3 of 5
Which inequality is true?
A. 1+ 7 < 10
B. 9/3n > 1
C. n-3> 1
D. 9n> 27