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Classify each of the following integrals as proper, improper and convergent, or improper and divergent. Evaluate any which are convergent if an indefinite integral can be found.

• [a [math]\int_0^2 \frac3{x^{\frac23}} dx[/math]
• [math]\int_1^2 \frac1{\sqrt{2-x}}dx[/math]
• [math]\int_2^3 \frac1{(x-2)^2} dx[/math]
• [math]\int_0^1 \frac1{x^2+x+1} dx[/math]
• [math]\int_0^{\frac{\pi}2} \tan x \; dx[/math]
• [math]\int_0^1 \frac{\sin x}x dx[/math].