Problem 4: Evaluate $\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$ by Algebraic Substitution

$\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$

$\qquad = \displaystyle \int \dfrac{y \, dy}{(1 + 2y)^{1/4}}$
 

Let
$u = (1 + 2y)^{1/4}$

$u^4 = 1 + 2y$

$y = \frac{1}{2}(u^4 - 1)$

$dy = 2u^3 \, du$

 

Hence,
$\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$

$\qquad = \displaystyle \int \dfrac{\frac{1}{2}(u^4 - 1)(2u^3 \, du)}{u}$

$\qquad = \displaystyle \int (u^4 - 1)(u^2 \, du)$

$\qquad = \displaystyle \int (u^6 - u^2) \, du$

$\qquad = \frac{1}{7}u^7 - \frac{1}{3}u^3 + C$

$\qquad = \frac{1}{21}u^3(3u^4 - 7) + C$
 

Revert to the original variable of integration
$\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$

$\qquad = \frac{1}{21}(1 + 2y)^{3/4}[ \, 3(1 + 2y) - 7 \, ] + C$

$\qquad = \frac{1}{21}(1 + 2y)^{3/4}(6y - 4) + C$

$\qquad = \frac{2}{21}(1 + 2y)^{3/4}(3y - 2) + C$   ←   answer