Question

If $[ t ]$ denotes the greatest integer $\leq t$, then the value of $\frac{3( e -1)}{ e } \int\limits_1^2 x^2 e^{[x]+\left[x^3\right]} dx$ is :

• A. $e^8-1$
• B. $e^7-1$
• C. $e^9-e$
• D. $e^8-e$

Answer 1

Answer: (D)

$e^8-e$

Answer 2

1∫2​x2e[x3]+1dx
x3=t
3x2dx=dt
=3e​1∫8​e[t]dt
=3e​{1∫2​edt+∫23​e2dt+……..+7∫8​e7dt}
=3e​(e+e2+………+e7)
=3e2​(1+e+………+e6)=3e2​(e−1)(e7−1)​
e3(e−1)​1∫2​x2×e[x]+[x3]dx=e3​(e−1)×3e2​(e−1)(e7−1)​ =e(e7−1)
=e8−e