Question:
If $I(x)=\int x^{2}(\log x)^{2} d x$ and $I( 1)=0$, then $I(x)$
If $I(x)=\int x^{2}(\log x)^{2} d x$ and $I( 1)=0$, then $I(x)$
Updated On: Jun 9, 2023
- $\frac{x^{3}}{18} \left[ 8\left(\log x\right)^{2} -3\log x\right] + \frac{7}{18} $
- $\frac{x^{3}}{27} \left[9\left(\log x\right)^{2} +6 \log x\right] - \frac{2}{27} $
- $\frac{x^{3}}{27} \left[9\left(\log x\right)^{2} - 6 \log x+2\right]- \frac{2}{27} $
- $\frac{x^{3}}{27} \left[9 \left(\log x\right)^{2} -6 \log x +2 \right] - \frac{2}{27}$
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The Correct Option is C
Solution and Explanation
Given integral
$I(x)=\int x^{2}(\log x)^{2} d x= \frac{x^{3}}{3}(\log x)^{2}-\int \frac{x^{3}}{3} \frac{2 \log x}{x} d x$
[by integration by parts]
$= \frac{x^{3}}{3}(\log x)^{2}-\frac{2}{3}\left[\frac{x^{3}}{3}(\log x)-\int \frac{x^{3}}{3}\left(\frac{1}{x}\right) d x\right]$
$= \frac{x^{3}}{3}(\log x)^{2}-\frac{2}{3}\left[\frac{x^{3}}{3}(\log x)-\frac{1}{3} \frac{x^{3}}{3}\right]+C$
$=\frac{x^{3}}{27}\left[9(\log x)^{2}-6(\log x)+2\right]+C$
$\because I(1)=0$
$\therefore \frac{2}{27}+C=0$
$\Rightarrow C=-\frac{2}{27}$
$\therefore I(x)=\frac{x^{3}}{27}\left[9(\log x)^{2}-6(\log x)+2\right]-\frac{2}{27}$
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Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.