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Differentiate your solution. Is the result you get equal to the integrand? If so, your work is correct.Anti Hydrogen said:the book shows a solution that i m not getting
https://es.symbolab.com/solver/integral-calculator/\int x^{2}\sqrt{a+bx}dxMath_QED said:Your book is right:
https://www.wolframalpha.com/input/?i=integral+u^2*(a+bu)^(1/2)+du
First mistake I saw in your solution is that you forgot the factor ##2## in the rule ##(a+b)^2 = a^2 + 2ab + b^2## while expanding ##((v-a)/b)^2##. I did not read further so it may contain further mistakes.
An antiderivative is a mathematical function that is the reverse of a derivative. It is used to find the original function when only the derivative is known.
Antiderivatives are useful in solving many real-world problems, such as finding the distance traveled by an object given its velocity or finding the original function of a rate of change.
One common trouble with finding an antiderivative is that there is not always a simple, closed-form solution. Another difficulty can arise when dealing with non-elementary functions, as they may not have a known antiderivative.
There are several techniques for finding antiderivatives, such as substitution, integration by parts, and partial fractions. It is also helpful to know common antiderivative formulas for basic functions.
Yes, an antiderivative and an indefinite integral refer to the same concept. The notation for an indefinite integral is ∫f(x)dx, while the notation for an antiderivative is F(x). Both represent the original function that has a given derivative.