Integrate the function x^2 x - Vidyadip

Question

Integrate the function $x^2 \log x$

✓

Answer

$\int {{x^2}\log xdx} $$= \int {\left( {\log x} \right){x^2}dx} $
$= \log x\int {{x^2}dx - \int {\left( {\frac{d}{{dx}}\log x\int {{x^2}dx} } \right)dx} } $
[Applying product rule]
$= \left( {\log x} \right)\frac{{{x^3}}}{3} - \int {\frac{1}{x}.\frac{{{x^3}}}{3}dx} $
$= \frac{{{x^3}}}{3}\log x - \frac{1}{3}\int{x^2}dx$
$= \frac{{{x^3}}}{3}\log x - \frac{1}{3}\frac{{{x^3}}}{3} + c$
$= \frac{{{x^3}}}{3}\log x - \frac{{{x^3}}}{9} + c$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks Question" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

$\text{Find} \lambda, \text{if the vectors} \overrightarrow{\text{a}} = \hat{\text{i}} + 3\hat{\text{j}} + \hat{\text{k}}, \overrightarrow{\text{b}} = 2\hat{\text{i}} - \hat{\text{j}} - \hat{\text{k}}$ $\text{and} \overrightarrow{\text{c}} = \lambda \hat{\text{j}} + 3\hat{\text{k}} $ $\text{are coplanar}.$

Evaluate the determinant $\left| {\begin{array}{*{20}{c}} 0&1&2 \\ { - 1}&0&{ - 3} \\ { - 2}&3&0 \end{array}} \right|$

A man of height 2 metres walks at a uniform speed of 5 km/h away from a lamp post which is 6 metres high. Find the rate at which the length of his shadow increases.

Find the maximum and minimum values, if any, of the function given by: $g(x) = x^3 + 1$

Find: $\int \frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}+4\right)} d x$

Find the value of $\mathrm{k}$ so that the function $\mathrm{f}$ is continuous at the indicated point: $ f(x)=\left\{\begin{array}{l} k x+1, \text { if } x \leq 5 \\ 3 x-5, \text { if } x>5 \end{array} \text { at } \mathrm{x}=5\right. $

Determine order and degree (if defined) of differential equations given in Exercise.
y" + 2y' + sin y = 0

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2}$

Write the number of all possible matrices of order $2\times3$ with each entry 1 or 2.

For the function $y = x^2$, if $x = 10$ and $\triangle\text{x}=0.1$. Find $\triangle\text{y}.$