Evaluate this integral of ydy/√y+1 using substitution method?

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Evaluate this integral of ydy/√y+1 using substitution method?

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Answer 1 · 2018-05-09T03:41:49

The integral equals $\frac{2}{3} {(y + 1)}^{\frac{3}{2}} - 2 {(y + 1)}^{\frac{1}{2}} + C$ $2/3(y + 1)^(3/2) - 2(y + 1)^(1/2) + C$

Explanation:

We have:

$I = \int \frac{y}{\sqrt{y + 1}} d y$ $I = int y/sqrt(y + 1) dy$

Now let $u = y + 1$ $u = y + 1$ . Then $d u = d y$ $du = dy$ .

$I = \int \frac{u - 1}{\sqrt{u}} d u$ $I = int (u - 1)/sqrt(u) du$

$I = \int \frac{u}{\sqrt{u}} - \frac{1}{\sqrt{u}} d u$ $I = int u/sqrt(u) - 1/sqrt(u)du$

$I = \int u^{\frac{1}{2}} - u^{- \frac{1}{2}} d u$ $I = int u^(1/2) - u^(-1/2)du$

$I = \frac{2}{3} u^{\frac{3}{2}} - 2 u^{\frac{1}{2}} + C$ $I = 2/3u^(3/2) - 2u^(1/2) + C$

$I = \frac{2}{3} {(y + 1)}^{\frac{3}{2}} - 2 {(y + 1)}^{\frac{1}{2}} + C$ $I = 2/3(y + 1)^(3/2) - 2(y + 1)^(1/2) + C$

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Evaluate this integral of ydy/√y+1 using substitution method?

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