Please help with the questions on definite integrals.

[Haru]

Hello, I am Haru and I have difficulties to solve the following problems. It would be much appreciated if someone can help me. Sorry for asking so many questions. And wish you Merry Christmas.

Consider the definite integral : ∫ ((dx)/((square root x)+(4x^(1/3)))) (which the lower bound (from): 0 and the upper bound (to): 1)
(a.) Then the most appropriate substitution to simplify this integral is u = ?
(b.) Then dx=f(x)du where f(x) = ?
(c.) After making the substitution and simplifying we obtain the integral ∫ g(u)du (which the lower bound (from): a and the upper bound (to): b) where
(i.) g(u)=
(ii.) a=
(iii.) b=
(iv.) This definite integral has value =

~ For 5 (a.) and (b), I think it require the use of Chain Rule, however, I do not know the best way to determine the most appropriate substitution and I do not know how to use the Chain Rule. Therefore, if would be much appreciated if ways or methods for determining the most appropriate substitution and Chain Rule can be provided.

~ For 5 (a.), my answer is ((square root x)+(4x^(1/3)))) as I think it is so complex that I should let u= ((square root x)+(4x^(1/3)))) so that I can differentiate easily, but it is wrong and I am confused. So I would like to know how to determine the most appropriate substitution.

Thank you.

 

[Deleted member 4993]

Hello, I am Haru and I have difficulties to solve the following problems. It would be much appreciated if someone can help me. Sorry for asking so many questions. And wish you Merry Christmas.

Consider the definite integral : ∫ ((dx)/((square root x)+(4x^(1/3)))) (which the lower bound (from): 0 and the upper bound (to): 1)
(a.) Then the most appropriate substitution to simplify this integral is u = ?
(b.) Then dx=f(x)du where f(x) = ?
(c.) After making the substitution and simplifying we obtain the integral ∫ g(u)du (which the lower bound (from): a and the upper bound (to): b) where
(i.) g(u)=
(ii.) a=
(iii.) b=
(iv.) This definite integral has value =

~ For 5 (a.) and (b), I think it require the use of Chain Rule, however, I do not know the best way to determine the most appropriate substitution and I do not know how to use the Chain Rule. Therefore, if would be much appreciated if ways or methods for determining the most appropriate substitution and Chain Rule can be provided.

~ For 5 (a.), my answer is ((square root x)+(4x^(1/3)))) as I think it is so complex that I should let u= ((square root x)+(4x^(1/3)))) so that I can differentiate easily, but it is wrong and I am confused. So I would like to know how to determine the most appropriate substitution.

Thank you.

(a) If you substitute (this probably is not the final answer) u = x^(1/6) ..........[ or.... u = x^1/6 ....or..... \(\displaystyle u = \sqrt[6]{x} \).... all are equivalent]

What do you get for dx ........and............. the integrand .................. and the limits?

To review "chain rule for integration", please investigate:

https://math.stackexchange.com/questions/1635949/is-there-a-chain-rule-for-integration/1635969