Integrating unit vector ρ

• I
When doing integration such as $\int_{0}^{2\pi} \hat{\rho} d\phi$ which would give us $2\pi \hat{\rho}$, must we decompose $\hat{ρ}$ into $sin(\phi) \hat{i} + cos(\phi) \hat{j}$ , then $\int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi$ , which would give us 0 instead?

Thanks

PeroK
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When doing integration such as $\int_{0}^{2\pi} \hat{\rho} d\phi$ which would give us $2\pi \hat{\rho}$, must we decompose $\hat{ρ}$ into $sin(\phi) \hat{i} + cos(\phi) \hat{j}$ , then $\int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi$ , which would give us 0 instead?

Thanks

Which method looks correct to you?

Which method looks correct to you?
The second one. My question is, is there anyway that I can keep using cylindrical coordinate without changing back to cartesian coordinate and get the same solution?

PeroK