Integration using various techniques

[physics604]

1. $$\int e^{x+e^x}\,dx$$

Homework Equations

Substitution, integration by parts

The Attempt at a Solution

$$u=e^x$$ $$\int e^{x+e^x}\,dx = \int e^x e^{e^x}\,dx = \int ue^u\,du$$
$$a=u$$ $$da=1du$$ $$dv=e^udu$$ $$v=e^u$$
$$=ue^u-\int e^u\,du = ue^u-e^u$$ $$=e^x e^{e^x}+e^{e^x} = e^{x+e^x}-e^{e^x} + C$$

The textbook's answer was $$e^{e^x} +C$$ How come my answer is different? Any help is much appreciated.

 

[Mark44]

1. $$\int e^{x+e^x}\,dx$$

Homework Equations

Substitution, integration by parts

The Attempt at a Solution

$$u=e^x$$

What is du? It is very important to also write du when you make a substitution u = <something>. In this case (and often), du is NOT equal to dx.

$$\int e^{x+e^x}\,dx = \int e^x e^{e^x}\,dx = \int ue^u\,du$$
$$a=u$$ $$da=1du$$ $$dv=e^udu$$ $$v=e^u$$
$$=ue^u-\int e^u\,du = ue^u-e^u$$ $$=e^x e^{e^x}+e^{e^x} = e^{x+e^x}-e^{e^x} + C$$

The textbook's answer was $$e^{e^x} +C$$ How come my answer is different? Any help is much appreciated.

 

[physics604]

$$du=e^xdx$$
How does this work into the calculation?

 

[haruspex]

$$du=e^xdx$$

But that's not what you did. You turned ##e^xdx## into ##udu##.

 

[physics604]

∫ex+exdx=∫exeexdx=∫ueudu

Do you mean I should do something like this? $$\int e^x\,dx = \int e^xe^{e^x}\,dx = \int ue^u\,e^x dx$$ But the point of my substitution was to turn everything into u's...

 

[Mark44]

You have ##\int e^x e^{e^x}dx##
u = e^x, so du = e^xdx

What does your integral look like using this substution?

 

[physics604]

Oh, I got it now. Thanks! My integral would be

$$\int e^u\,du$$

 

[Mark44]

Yes.