Answer:
Solution : 3
Step-by-step explanation:
Well there are actually two approaches to this. (1) You could apply the power rule, making it a bit simpler, or (2) use the approach given. Let's just use the approach given so you can learn it as assigned that way:
[tex]\lim _{t\to 0^+}\int _t^1x^{-\frac{2}{3}}\:\\[/tex]
Funny thing is we will use the power rule anyhow when solving this problem. We want to start by evaluating x^-2/3 on the interval [t to 1].
[tex]\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1}\\=> \left[\frac{x^{-\frac{2}{3}+1}}{-\frac{2}{3}+1}\right]^1_t\\\\=> \left[3x^{\frac{1}{3}}\right]^1_t\\\\\mathrm{Compute\:the\:boundaries}\\=> 3-3t^{\frac{1}{3}}[/tex]
And now you know that we would have to substitute this value, 3 - 3t^1/3, back into the primary expression.
[tex]\lim _{t\to \:0+}\left(3-3t^{\frac{1}{3}}\right),\\\\\mathrm{Plug\:in\:the\:value}\:t=0\\=> 3-3\cdot \:0^{\frac{1}{3}}\\=> 3[/tex]
