Troubles with an antiderivative

[Anti Hydrogen]

Homework Statement:

the book shows a solution that i m not getting

Relevant Equations:

i attached the file (word file) please check it out tell me if i am right or wrong

please tell me if i am right or wrong

 

[Mark44]

the book shows a solution that i m not getting

Differentiate your solution. Is the result you get equal to the integrand? If so, your work is correct.
Differentiate the book's solution. Is the result equal to the integrand? If so, the book's solution is correct.

Note that it is possible for two people to get two different antiderivatives, with both being correct. In such cases, the two antiderivatives differ by a constant.

 

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 $\left(\frac{v-a}{b}\right)^2$. I did not read further so it may contain further mistakes.

 

[Anti Hydrogen]

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.

https://es.symbolab.com/solver/integral-calculator/\int x^{2}\sqrt{a+bx}dx