Integrate the function: \(x^2 log\ x\)
Solution and Explanation
Let \( I\) =\(∫\)\(x^2 log\ x\ dx\)
Taking \(log\ x\) as first function and \(x^2\) as second function and integrating by parts, we obtain
\(I= log\ x∫[x^2dx-∫{(\frac {d}{dx}log\ x)∫x^2dx]}dx\)
\(I= log\ x(\frac {x^3}{3})-∫\frac 1x.\frac {x^3}{3}dx\)
\(I=\frac { x^3log\ x}{3}-∫\frac {x^2}{3}dx\)
\(I= \frac {x^3log\ x}{3}-\frac {x^3}{9}+C\)
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Let \( f : (0, \infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0 \), } \[ \int_0^a f(x) \, dx = f(a), \quad f(1) = 1, \quad f(16) = \frac{1}{8}, \quad \text{then } 16 - f^{-1}\left( \frac{1}{16} \right) \text{ is equal to:}\]
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\[ \text{Sn}(s) \mid \text{Sn}^{2+} (\text{0.004 M}) \parallel \text{H}^+ (\text{0.02 M}) \mid \text{H}_2 (\text{1 Bar}) \mid \text{Pt}(s) \]
(Given: \( E^\circ_{\text{Sn}^{2+}/\text{Sn}} = -0.14 \, \text{V}, E^\circ_{\text{H}^+/\text{H}_2} = 0.00 \, \text{V} \))- CBSE CLASS XII - 2025
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If vector \( \mathbf{a} = 3 \hat{i} + 2 \hat{j} - \hat{k} \) \text{ and } \( \mathbf{b} = \hat{i} - \hat{j} + \hat{k} \), then which of the following is correct?
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(a) Calculate the amount of work done to separate the two charges at infinite distance.
(b) If this system of charges was initially kept in an electric field \[ \vec{E} = \frac{A}{r^2}, \text{ where } A = 8 \times 10^4\, \text{N}\,\text{C}^{-1}\,\text{m}^2, \] calculate the electrostatic potential energy of the system.- CBSE CLASS XII - 2025
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Concepts Used:
Integration by Partial Fractions
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
