How do you integrate $\int x \sqrt{x - 1}$ $int xsqrt(x-1)$ by parts?

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How do you integrate $\int x \sqrt{x - 1}$ $int xsqrt(x-1)$ by parts?

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Answer 1 · 2017-01-30T15:16:46

$x = {cos}^{2} y$ $x = cos^2 y$
$\sqrt{x - 1} = sin y$ $sqrt(x-1) = siny$
$I = \int {cos}^{2} y sin y 2 cos y sin y (- 1)$ $I = int cos^2 y siny 2cosy siny (-1)$
$I = \int (- 2) {cos}^{3} y sin y$ $I = int (-2) cos^3ysiny$
$I = \frac{1}{2} {cos}^{4} y$ $I = 1/2 cos^4 y$

Answer 2 · 2017-01-30T18:08:41

Please see below.

Explanation:

$\int x \sqrt{x - 1} d x$ $intxsqrt(x-1) dx$

Let $u = x$ $u = x$ and $d v = \sqrt{x - 1} d x$ $dv = sqrt(x-1) dx$

so that $d u = 1 d x$ $du=1dx$ and $v = \frac{2}{3} {(x - 1)}^{\frac{3}{2}}$ $v = 2/3(x-1)^(3/2)$

$u v - \int v d u = \frac{2}{3} x {(x - 1)}^{\frac{3}{2}} - \frac{2}{3} \int {(x - 1)}^{\frac{3}{2}} d x$ $uv-intvdu = 2/3x(x-1)^(3/2) - 2/3 int (x-1)^(3/2) dx$

$= \frac{2}{3} x {(x - 1)}^{\frac{3}{2}} - \frac{2}{3} [\frac{2}{5} {(x - 1)}^{\frac{5}{2}}] + C$ $= 2/3x(x-1)^(3/2) - 2/3 [2/5 (x-1)^(5/2)] +C$

$= \frac{2}{3} x {(x - 1)}^{\frac{3}{2}} - \frac{4}{15} {(x - 1)}^{\frac{5}{2}} + C$ $= 2/3x(x-1)^(3/2) - 4/15 (x-1)^(5/2) +C$ .

Rewrite algebraically to taste. I like the answer above, but others might prefer

$= \frac{2}{15} [5 x {(x - 1)}^{\frac{3}{2}} - 2 {(x - 1)}^{\frac{5}{2}}] + C$ $= 2/15[5x(x-1)^(3/2) - 2 (x-1)^(5/2)]+C$

Or

$= \frac{2}{15} {(x - 1)}^{\frac{3}{2}} (3 x + 2) + C$ $= 2/15(x-1)^(3/2)(3x+2)+C$

Or some equivalent expression.

Answer 3 · 2017-01-30T19:29:59

Without integration by parts one can see that
$x = x - 1 + 1$ $x = x - 1 + 1$
so
$x \sqrt{x - 1} = (x - 1) \sqrt{x - 1} + \sqrt{x - 1}$ $x sqrt(x-1) = (x-1)sqrt(x-1) + sqrt(x-1)$
$\int x \sqrt{x - 1} d x = \int {(x - 1)}^{\frac{3}{2}} d (x - 1) + \int {(x - 1)}^{\frac{1}{2}} d (x - 1)$ $int x sqrt(x-1) dx = int (x-1)^(3/2)d(x-1) + int (x-1)^(1/2)d(x-1)$
$\int x \sqrt{x - 1} d x = {(x - 1)}^{\frac{5}{2}} \frac{2}{5} + {(x - 1)}^{\frac{3}{2}} \frac{2}{3}$ $int x sqrt(x-1) dx = (x-1)^(5/2)2/5 + (x-1)^(3/2)2/3$

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How do you integrate $\int x \sqrt{x - 1}$ $int xsqrt(x-1)$ by parts?

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