Integrate the function (x^2 + 1) x - Vidyadip

Question

Integrate the function $(x^2 + 1) \log x$

✓

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 Question" from Integrals→Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

$\text{If} f(x) = x + 7 \text{and g (x)} = x - 7, x \in \text{R, find (fog) (7)} $

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi vertical angle is $\tan^{-1}(0.5)$ water is poured into it at a constant rate of $5$ cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is $4 \ m$.

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that the youngest is a girl?

Find $\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$

Show that the function given by $f(x) = 7x – 3$ is increasing on $R.$

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that one of them is black and other is red.

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^2\text{y}}{\text{dx}^2}+5\text{x}\Big(\frac{\text{dy}}{\text{dx}}\Big)-6\text{y}=\log\text{x}$

Find the principal value of $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Integrate the functions in Exercises:
$\frac{1}{\text{x}+\text{x}\log\text{x}}$

The area of the region bounded by the circle $x^2 + y^2 = 1$ is$:$