Mathematica Asked by A little mouse on the pampas on December 4, 2020

I want to expand the following two functions into series at `x = 0`

, but MMA(Version 12.1.1) runs all the time and cannot return results:

```
Series[Integrate[Log[1 + Sqrt[t^3]], {t, 0, x}], {x, 0, 5}]
Series[Integrate[Sqrt[Sin[t]^3], {t, 0, 1 - Cos[x]}], {x, 0, 5},
Assumptions -> (x > 0)]
```

What can I do to find their series expansion quickly?

You can use the new in M11.3 function `AsymptoticIntegrate`

for this purpose:

```
AsymptoticIntegrate[Log[1+Sqrt[t^3]], {t, 0, x}, {x, 0, 5}]
AsymptoticIntegrate[Sqrt[Sin[t]^3], {t, 0, x}, {x, 0, 5}]
```

-(x^4/8) + (2 x Sqrt[x^3])/5 + 2/33 x^4 Sqrt[x^3]

(2 x Sqrt[x^3])/5 - 1/18 (x^3)^(3/2)

Correct answer by Carl Woll on December 4, 2020

Include `Assumptions->x>0`

inside `Integrate`

```
Series[Integrate[Log[1 + Sqrt[t^3]], {t, 0, x}, Assumptions -> x > 0], {x, 0, 5}]
(*(2 x^(5/2))/5 - x^4/8*)
Series[Integrate[Sqrt[Sin[t]^3], {t, 0, x}, Assumptions -> (x > 0)], {x, 0, 5}]
(*(2 x^(5/2))/5 - x^(9/2)/18*)
```

Answered by Ulrich Neumann on December 4, 2020

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