Question

How do you do this question?

Answer 1

Solution : 3

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:

`\lim _(t\to 0^+)\int _t^1x^{-(2)/(3)}\:\\`

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].

`\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=(x^(a+1))/(a+1)\\=> \left[\frac{x^{-(2)/(3)+1}}{-(2)/(3)+1}\right]^1_t\\\\=> \left[3x^{(1)/(3)}\right]^1_t\\\\\mathrm{Compute\:the\:boundaries}\\=> 3-3t^{(1)/(3)}`

And now you know that we would have to substitute this value, 3 - 3t^1/3, back into the primary expression.

`\lim _(t\to \:0+)\left(3-3t^{(1)/(3)}\right),\\\\\mathrm{Plug\:in\:the\:value}\:t=0\\=> 3-3\cdot \:0^{(1)/(3)}\\=> 3`