Integrate WITHOUT using Integration by Parts

[tkvictim]

Integrate this indefinite integral without using integration by parts.
Show all work. [and probably, by not using the calculator either]

Is that even possible?

10te[sup:qb9i76on]-t/3[/sup:qb9i76on]

 

[BigGlenntheHeavy]

$\displaystyle Hey, \ bottomless \ pit, \ do \ us \ and \ yourself \ a \ favor \ and \ hit \ the \ bricks.$

 

[Deleted member 4993]

Integrate this indefinite integral without using integration by parts.
Show all work. [and probably, by not using the calculator either]

Is that even possible?

10te[sup:h7gxruom]-t/3[/sup:h7gxruom]

Yes it is possible.

Do you know how to expand 10te[sup:h7gxruom]-t/3[/sup:h7gxruom] as a polynomial function of 't'?

.

 

[BigGlenntheHeavy]

$\displaystyle 10te^{-t/3} \ = \ 10 \sum_{n=0}^{\infty}\frac{(-1)^n t^{n+1}}{3^{n}n!} \ converges \ for \ all \ t \ (Maclauren \ Series).$

$\displaystyle \int [10te^{-t/3}]dt \ = \ \int\bigg[10 \sum_{n=0}^{\infty}\frac{(-1)^n t^{n+1}}{3^{n}n!} \bigg]dt$

$\displaystyle Hence, \ \int [10te^{-t/3}]dt \ = \ 10 \sum_{n=0}^{\infty}\frac{(-1)^n t^{n+2}}{3^{n}n!(n+2)}$

$\displaystyle Check: \ \int_{0}^{15}[10te^{-t/3}]dt \ = \ 10 \sum_{n=0}^{\infty}\frac{(-1)^n 15^{n+2}}{3^{n}n!(n+2)} \ \dot= \ 86.3615086205$

$\displaystyle However, \ since \ "the \ bottomless \ pit" \ doesn't \ understand \ I \ by \ P, \ we \ are \ obviously \ dealing \ with$

$\displaystyle \ a \ fisherman.$