Question:
If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\int\limits^{{2}}_{{0}}x^2 [x] dx$
If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\int\limits^{{2}}_{{0}}x^2 [x] dx$
Updated On: Feb 15, 2024
- $\frac{5}{3}$
- $\frac{7}{3}$
- $\frac{8}{3}$
- $\frac{4}{3}$
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The Correct Option is B
Solution and Explanation
$\int\limits^{{2}}_{{0}}x^2 [x] \cdot dx=\int\limits^{{1}}_{{0}}x^2\times 0dx+\int\limits^{{2}}_{{1}}x^2 \times 1dx=(x^3/3)^2_1=7/3$
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
