Question

The greatest integer less than or equal to


`\rm \int_(1)^2 log_(2)( {x}^(3) + 1 ) dx + \int_( 1 )^{log_(2)(9)} ( {2}^x - 1) {}^{ (1)/(3) } \: dx \: is \\`
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Answer 1

We use a property from an earlier question of yours [27174924], that

`\displaystyle \int_a^b f(x) \, dx + \int_(f(a))^(f(b)) f^(-1)(x) \, dx = b\,f(b) - a\, f(a)`

Note that

`f(x) = \log_2(x^3+1) \implies f^(-1)(x) = \left(2^x - 1\right)^(1/3)`

`a = 1 \implies f(a) = \log_2(1^3 + 1) = \log_2(2) = 1`

`b = 2 \implies f(b) = \log_2(2^3 + 1) = \log_2(9)`

so that the exact value of the integral is

`\displaystyle \int_1^2 \log_2(x^3 + 1) \, dx + \int_1^(\log_2(9)) \left(2^x - 1\right)^(1/3) = 2\log_2(9) - 1`

Now, observe that

`8 = 2^3 < 9 < 2^(7/2) = 8\sqrt2 \approx11.3 \\\\ \implies 3 < \log_2(9) < \frac72 \\\\ \implies 6 < 2\log_2(9) < 7 \\\\ \implies 5 < 2\log_2(9) - 1 < 6`

so that

`\left\lfloor 2\log_2(9) - 1 \right\rfloor = \boxed{5}`