1 | initial version |

Both answers are correct.

To see it by hand, we can use:

- $(x + 1)^{5/2} = (x + 1) (x + 1)^{3/2} = x (x + 1)^{3/2} + (x + 1)^{3/2}$
- $(x + 1)^{3/2} = (x + 1) (x + 1)^{1/2} = x (x + 1)^{1/2} + (x + 1)^{1/2}$

to express both as linear combinations of `(x + 1)^(1/2)`

and `x * (x + 1)^(1/2)`

.

To see it with Sage, we can differentiate both and compare them.

Define the function to integrate:

```
sage: x = var('x')
sage: f = x*sqrt(x + 1)
sage: f
sqrt(x + 1)*x
```

Integrate with Sage:

```
sage: F = integral(f, x)
sage: F
2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)
```

The primitive you computed (up to a constant):

```
sage: G = (2/3)*x*(x + 1)^(3/2) - (4/15)*(x + 1)^(5/2)
sage: G
-4/15*(x + 1)^(5/2) + 2/3*(x + 1)^(3/2)*x
```

To see that they are both correct:

```
sage: bool(F == G)
True
```

Or check they have the same derivative:

```
sage: g = G.diff(x)
sage: g
sqrt(x + 1)*x
sage: ff = F.diff(x)
sage: ff
(x + 1)^(3/2) - sqrt(x + 1)
sage: bool(g == ff)
True
```

Or more visually, compare the plots of `F`

and `G`

(or `f`

and `ff`

):

```
sage: pF = plot(F, (-1, 1.6), color='firebrick')
sage: pG = plot(G, (-1, 1.6), color='steelblue')
sage: p = graphics_array([pF, pG, pF + pG])
sage: p.show(figsize=(7, 3))
Launched png viewer for Graphics Array of size 1 x 3
```

2 | No.2 Revision |

Both answers are correct.

To see it by hand, we can use:

- $(x + 1)^{5/2} = (x + 1) (x + 1)^{3/2} = x (x + 1)^{3/2} + (x + 1)^{3/2}$
- $(x + 1)^{3/2} = (x + 1) (x + 1)^{1/2} = x (x + 1)^{1/2} + (x + 1)^{1/2}$

to express both as linear combinations of $(x + 1)^{1/2}$,
$x (x + 1)^1/2}$ and `(x + 1)^(1/2)`

$x^2 (x + 1)^{1/2}$.`x * (x + 1)^(1/2)`

.

To see it with Sage, we can differentiate both and compare them.

Define the function to integrate:

```
sage: x = var('x')
sage: f = x*sqrt(x + 1)
sage: f
sqrt(x + 1)*x
```

~~Integrate ~~Compute a primitive (aka antiderivative) with Sage:

```
sage: F = integral(f, x)
sage: F
2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)
```

The primitive you computed (up to a constant):

```
sage: G = (2/3)*x*(x + 1)^(3/2) - (4/15)*(x + 1)^(5/2)
sage: G
-4/15*(x + 1)^(5/2) + 2/3*(x + 1)^(3/2)*x
```

~~To see ~~Check that they ~~are both correct:~~agree:

```
sage: bool(F == G)
True
```

~~Or check ~~Check they have the same derivative:

```
sage: g = G.diff(x)
sage: g
sqrt(x + 1)*x
sage: ff = F.diff(x)
sage: ff
(x + 1)^(3/2) - sqrt(x + 1)
sage: bool(g == ff)
True
```

Or more visually, compare the plots of `F`

and `G`

(or `f`

and `ff`

):

```
sage: pF = plot(F, (-1, 1.6), color='firebrick')
sage: pG = plot(G, (-1, 1.6), color='steelblue')
sage: p = graphics_array([pF, pG, pF + pG])
sage: p.show(figsize=(7, 3))
Launched png viewer for Graphics Array of size 1 x 3
```

3 | No.3 Revision |

Both answers are correct.

To see it by hand, we can use:

- $(x + 1)^{5/2} = (x + 1) (x + 1)^{3/2} = x (x + 1)^{3/2} + (x + 1)^{3/2}$
- $(x + 1)^{3/2} = (x + 1) (x + 1)^{1/2} = x (x + 1)^{1/2} + (x + 1)^{1/2}$

to express both as linear combinations of $(x + 1)^{1/2}$,
$x (x + ~~1)^1/2}$ ~~1)^{1/2}$ and $x^2 (x + 1)^{1/2}$.

To see it with Sage, we ~~can ~~can

- ask Sage whether the two different expressions in fact agree
- differentiate both and compare
~~them.~~them - plot the functions to visually check whether they agree

Define the function to integrate:

```
sage: x = var('x')
sage: f = x*sqrt(x + 1)
sage: f
sqrt(x + 1)*x
```

Compute a primitive (aka antiderivative) with Sage:

```
sage: F = integral(f, x)
sage: F
2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)
```

The primitive you computed (up to a constant):

```
sage: G = (2/3)*x*(x + 1)^(3/2) - (4/15)*(x + 1)^(5/2)
sage: G
-4/15*(x + 1)^(5/2) + 2/3*(x + 1)^(3/2)*x
```

Check that they agree:

```
sage: bool(F == G)
True
```

Check they have the same derivative:

```
sage: g = G.diff(x)
sage: g
sqrt(x + 1)*x
sage: ff = F.diff(x)
sage: ff
(x + 1)^(3/2) - sqrt(x + 1)
sage: bool(g == ff)
True
```

Or more visually, compare the plots of `F`

and `G`

(or `f`

and `ff`

):

```
sage: pF = plot(F, (-1, 1.6), color='firebrick')
sage: pG = plot(G, (-1, 1.6), color='steelblue')
sage: p = graphics_array([pF, pG, pF + pG])
sage: p.show(figsize=(7, 3))
Launched png viewer for Graphics Array of size 1 x 3
```

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