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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
The greatest integer less than or equal to $\int_1^2 \log _2\left(x^3+1\right) d x+\int_1^{\log 2}\left(2^x-1\right)^{\frac{1}{3}} d x$ is. . . . . 
  • A
    $2$
  • B
    $3$
  • C
    $4$
  • ✓
    $5$
✓

Answer

Correct option: D.
$5$
d
$f(x)=\log _2\left(x^3+1\right)=y$

$x^3+1=2^y \Rightarrow x=\left(2^y-1\right)^{1 / 3}=f^{-1}(y)$

$f^{-1}(x)=\left(2^x-1\right)^{1 / 3}$

$=\int_1^2 \log _2\left(x^3+1\right) d x+\int_1^{\log _2 9}\left(2^x-1\right)^{1 / 3} d x$

$=\int_1^2 f(x) d x+\int_1^{\log _2 9} f^{-1}(x) d x=2 \log _2 9-1$

$=8<9<2^{7 / 2} \Rightarrow 3<\log _2 9<\frac{7}{2}$

$=5<2 \log _2 9-1<6$

${\left[2 \log _2 9-1\right]=5}$

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