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physics604
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1. $$\int e^{x+e^x}\,dx$$
Substitution, integration by parts
$$u=e^x$$ $$\int e^{x+e^x}\,dx = \int e^x e^{e^x}\,dx = \int ue^u\,du$$
$$a=u$$ $$da=1du$$ $$dv=e^udu$$ $$v=e^u$$
$$=ue^u-\int e^u\,du = ue^u-e^u$$ $$=e^x e^{e^x}+e^{e^x} = e^{x+e^x}-e^{e^x} + C$$
The textbook's answer was $$e^{e^x} +C$$ How come my answer is different? Any help is much appreciated.
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.