If [x] denotes the greatest integer function, then ₀^{5} x²[x] d x=

Correct Answer is : 4003400/33400

Correct Answer is : 4003400/33400

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MHTCET 16 May 2024 Shift 2 Paper

Question : 126 of 150

Marks: +1, -0

[x]

\int_{0}^{5} x^{2}[x] d x=

Solution:

Concept:

The greatest integer function

[x]

is constant on each interval

[n, n+1)

, so the integral splits into a sum of intervals with

[x] = n

Explanation:

Split the integral from

0

5

at integer points:

\int_{0}^{5} x^{2}[x]\,dx=\sum_{n=0}^{4}\int_{n}^{n+1} x^{2}\cdot n\,dx.

Since

[x]=0

for

0 \le x < 1

, the first term is zero.

For

n=1,2,3,4

, evaluate each term:

\int_{1}^{2}x^{2}\cdot1\,dx=\left[\frac{x^{3}}{3}\right]_{1}^{2}=\frac{8-1}{3}=\frac{7}{3}.

\int_{2}^{3}x^{2}\cdot2\,dx=2\cdot\frac{27-8}{3}=\frac{38}{3}.

\int_{3}^{4}x^{2}\cdot3\,dx=3\cdot\frac{64-27}{3}=37.

\int_{4}^{5}x^{2}\cdot4\,dx=4\cdot\frac{125-64}{3}=\frac{244}{3}.

Adding these:

\frac{7}{3}+\frac{38}{3}+37+\frac{244}{3}=\frac{400}{3}.

Answer:

\int_{0}^{5} x^{2}[x]\,dx=\frac{400}{3}.

Correct option: D.

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