Integrate the function (x2 + 1) log x - Vidyadip

Question

Integrate the function (x² + 1) log x

✓

Answer

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

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" 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

If the function $\text{f(x)}=\frac{\sin10\text{x}}{\text{x}},\text{ x}\neq0$ is continuous at x = 0, find f(0).

If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?

Write the degree of the differential equation $\text{a}^{2}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\left\{1+(\frac{\text{dy}}{\text{dx}})^{2}\right\}^{\frac{1}{4}}.$

Find the value of ‘p’ for which the vectors $3\hat{\text{i}} + 2\hat{\text{j}} + 9 \hat{\text{k}}a\text{ and } \hat{\text{i}} - 2\text{p} \hat{\text{j}} + 3\hat{\text{k}}\text{ are parallel. }$

Write the position vector of the point which divides the join of points with position vectors $3\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}} \text{and } 2\overrightarrow{\text{a}} + 3\overrightarrow{\text{b}}$ in the ratio $2:1.$

Evaluate the following:
$\text{cosec}^{-1}\Big\{\text{cosec}\Big(-\frac{9\pi}{4}\Big)\Big\}$

Integrate the functions in Exercises:
$\frac{\cos\text{x}}{\sqrt{1+\sin\text{x}}}$

If $\theta $ is the angle between two vectors$\;\vec a\;$and $\vec b$, then $\vec a.\vec b \geq 0$ only when

Consider the experiment of tossing a coin. If the coin shows head, toss it again, but if it shows tail, then throw a die. Find the conditional probability of the event that the die shows a number greater than 4, given that there is atleast one tail.

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