Solving an Anti-Derivative Problem with Trigonometric Identities

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In summary, the anti-derivative of f(x) = 4 - 3(1+x^2)^(-1) can be found by taking the anti-derivative of each term individually. The anti-derivative of 4 is 4x. For the term 3(1+x^2)^(-1), the anti-derivative can be found using the identity of the derivative of arctan(x). Therefore, the final answer is 4x - 3 arctan(x) + C. The proof of this identity can be found in various sources such as textbooks or online resources.
  • #1
zhen
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find the anti-derivative of :

f(x) = 4 - 3(1+x^2)^(-1)

I have thought this question for hours...but no clue at all...

that is what I have attempted:

F(x) = 4x - 3 Ln(1 + x^2) ...
but if i differentiated it ---
then I got F'(x) = 4 - 3*(2x)/(1 + x^2)...

is there anyway to eliminate the (2x) ...?

please help
 
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  • #2
Well, first the "4" is obvious- you are completely correct that it's anti-derivative is 4x.

Now, for 3/(1+ x^2). Log doesn't work because 1+ x^2 is not x!

Do you know the derivative of arctan(x)?
 
  • #3
oh...thank you...
I totally forgot about there are some formula for that...
yes...
so is the answer 4x - 3 arc tan x + C?
 
  • #4
but in my textbook, i can not find the prove of those identities...
just wonder if there is any link for that...
 

FAQ: Solving an Anti-Derivative Problem with Trigonometric Identities

What is an anti-derivative?

An anti-derivative, also known as an indefinite integral, is a function that, when differentiated, gives the original function. In other words, it is the reverse process of differentiation.

Why is solving anti-derivative problems important?

Solving anti-derivative problems is important because it allows us to find the original function when we are given its derivative. This is useful in many areas of science, such as physics and economics, where we need to find the position or velocity of an object based on its acceleration.

What are the techniques for finding anti-derivatives?

The most common techniques for finding anti-derivatives are the power rule, integration by substitution, integration by parts, and partial fractions. These techniques involve manipulating the original function in order to find a function whose derivative is the original function.

What is the difference between a definite and indefinite integral?

A definite integral has specific limits of integration, whereas an indefinite integral does not. This means that a definite integral will give a numerical value, while an indefinite integral will give a function. In other words, a definite integral gives us the area under the curve of a function, while an indefinite integral gives us the original function itself.

How can I check my answer when solving an anti-derivative problem?

You can check your answer by taking the derivative of your anti-derivative and seeing if it matches the original function. If it does, then your solution is correct. You can also use online tools or graphing calculators to graph both the original function and your anti-derivative and see if they match.

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