What is the problem with this method while integrating $(e^x-(2x+3)^4)^3$?

Mathematics Asked on January 3, 2022

What is the problem with this method while integrating $$(e^x-(2x+3)^4)^3$$?

I already know what is its integration. I collected the answers from Quora (black ones) and WolframAlpha website (the red one).

Alright, then I tried to solve the integration in this method. But, the answer appeared different. The mathematical approach seemed very legitimate to me. Where did I go wrong, would you kindly point that out?

now here is the difference between my itegrals(1) graph and the graph that gave wolphrapalpha(2).

I used same method to solve this two functions i found. and it worked.


BUT WHY IT DID NOT WORK FOR THAT ONE, WHERE DID I WENT WRONG?


Your methodology is highly nonstandard and suspicious, but I will try to formalize it.

You appear to deal with integrals of the form $$int f(x)^n dx$$

I think your general strategy is

1. Set $$y= f(x)$$

2. Integrate $$y^n$$ to get $$y^{n+1}/(n+1)$$.

3. Divide $$y^{n+1}/(n+1)$$ by $$dy$$ and then write everything in terms of $$x$$.

So in the end you are saying that the following is an antiderivative for $$f(x)^n$$ $$frac{f(x)^{n+1}}{(n+1) f'(x)}tag{1}$$

So let's take the derivative of the last function and see what we get (we should get $$f(x)^n$$ if your strategy is correct). Applying the quotient rule and chain rule, we get the derivative of the function in $$(1)$$ as:

$$frac{f(x)^n (f'(x))^2-frac{1}{n+1}f(x)^{n+1} f''(x)}{(f'(x))^2}tag{2}$$

In general, there is no reason to expect that the function in $$(2)$$ is going to equal $$f(x)^n$$.

On the other hand, if we are in the special case that $$f'(x)$$ is a nonzero constant, then the expression in $$(2)$$ is equal to $$f(x)^n$$ since in this case $$f''(x)=0$$. This is precisely what happens in your two easier examples where the strategy works. But this is not the case in the more complicated example.

Answered by halrankard on January 3, 2022

Related Questions

Does the Riemannian distance function obey the Leibniz rule?

1  Asked on November 24, 2021 by asaf-shachar

Non Case Analysis Proof of Triangle Inequality for Hamming Distance

1  Asked on November 24, 2021 by henry-powell

A non-circular argument that uses the Maclaurin series expansions of $sin x$ and $cos x$ to show that $frac{d}{dx}sin x = cos x$

1  Asked on November 24, 2021

$|int_a^b f(x)dx| leq |int_a^b |f(x)| dx|$?

1  Asked on November 24, 2021

Number of permutations of the letters $a, b, c, d$ such that $b$ does not follow $a$, and $c$ does not follow $b$, and $d$ does not follow $c$

2  Asked on November 24, 2021

Diagonalizability of an operator

1  Asked on November 24, 2021 by gba

What is the probability for $x$ to be positive only?

2  Asked on November 24, 2021 by amateurashish

A simple proof of Sylow theorem for abelian groups

1  Asked on November 24, 2021

Question about the proof of the index theorem appearing in Milnor’s Morse Theory

1  Asked on November 24, 2021 by doggerel

Why can 2 uncorrelated random variables be dependent?

3  Asked on November 24, 2021

How to solve $intfrac{1}{sqrt {2x} – sqrt {x+4}} , mathrm{dx}$?

3  Asked on November 24, 2021 by flinn-bella

Integral: $int dfrac{dx}{(x^2-4x+13)^2}$?

9  Asked on November 24, 2021 by user809843

What is the intersection of inductive definable subsets of a real closed field?

2  Asked on November 24, 2021

How can I prove that the following are happening: $lnBig(1+frac{1}{x}Big)=frac{1}{x}+oBig(frac{1}{x}Big)$?

2  Asked on November 24, 2021 by andvld

Having (x,y,t) determine if two persons were near for more than 5 minutes

1  Asked on November 24, 2021

Understanding notation and meaning of uniform convergence of a power series

1  Asked on November 24, 2021

Proving L’Hospital’s rule

1  Asked on November 24, 2021 by blackthunder

Is it possible to calculate $x$ and $y$?

1  Asked on November 24, 2021

Sufficient conditions for ideal to be in kernel of ring homomorphism

2  Asked on November 24, 2021 by cloud-jr-k

Each permutation in permutation group PSL2 consist of fixed points and cycles of equal lengths. Prove or disprove it.

1  Asked on November 24, 2021 by slepecky-mamut