If [x] denotes the greatest integer x, then the value of _1^2 x^2,e^[x]+[x^3],dx is:

Question

If $[x]$ denotes the greatest integer $\le x$, then the value of $\displaystyle\int_1^2 x^2\,e^{[x]+[x^3]}\,dx$ is:

Accepted Answer

On $(1,2)$, $[x]=1$. Let $x^3=t$, $3x^2dx=dt$:

$$I = \int_1^2 x^2e^{1+[x^3]}\,dx = \frac{e}3\int_1^8 e^{[t]}\,dt = \frac{e}3\left(\int_1^2e\,dt+\int_2^3e^2\,dt+\cdots+\int_7^8e^7\,dt\right)$$

$$= \frac{e}3\left(e+e^2+\cdots+e^7\right) = \frac{e^2}3(1+e+\cdots+e^6) = \frac{e^2}3\cdot\frac{e^7-1}{e-1}$$

$$I = \frac{e^2}{3}\cdot\frac{(e-1)(e^6+e^5+\cdots+1)}{e-1}=\frac{e}{3}(e-1)\cdot\frac{e^7-1}{e-1} = e(e^7-1) = e^8-e$$

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