Evaluate the following definite integrals: _ 2 ^3 x x^2+1 dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{2}^\limits{3}\frac{\text{x}}{\text{x}^2+1} \text{ dx}$

✓

Answer

Let $\text{I}=\int_{2}^\limits{3}\frac{\text{x}}{\text{x}^2+1} \text{ dx}$ Then,
$\text{I}=\frac{1}{2}\int_{2}^\limits{3}\frac{2\text{x}}{\text{x}^2+1} $
$\Rightarrow\text{I}=\frac{1}{2}\big[\log(\text{x}^2-1)\big]^3_2$
$\Rightarrow\text{I}=\frac{1}{2}\big(\log10-\log5\big)$
$\Rightarrow\text{I}=\frac{1}{2}\log\frac{10}{5}$ $\Big[\because\log\text{a}-\log\text{b}=\log\frac{\text{a}}{\text{b}}\Big]$
$\Rightarrow\text{I}=\frac{1}{2}\log2$

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

Similar questions

Write the derivative of $f(x) = |x|^3$ at $x = 0.$

Urn $A$ contains $1$ white $, 2$ black and $3$ red balls; urn $B$ contains $2$ white $,1$ black and $1$ red ball; and urn $C$ contains $4$ white $,5$ black and $3$ red balls. One urn is chosen at random and two balls are drawn. These happen to be one white and one red. What is the probability that they come from urn $A$?

Write a value of $\int(\text{e}^{\text{x}\log_\text{e}\text{a}}+\text{e}^{\text{a}\log_\text{e}\text{x}})\text{dx}$

If x and y are connected parametrically by the equations given in Exercise without eliminating the parameter, Find $\frac{\text{dy}}{\text{dx}}.$
$\text{x}=\text{a}\sec\theta,\text{y}=\text{b}\tan\theta$

Given the matrices
$\text{A}=\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4\end{bmatrix},\text{B}=\begin{bmatrix}9&7&-1\\3&5&4\\2&1&6\end{bmatrix}$ and $\text{C}=\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix}$ Verify that (A + B) + C = A + (B + C).

If $\vec{\text{a}}$ are $\vec{\text{b}}$ are two unit vectors such that $\vec{\text{a}}+\vec{\text{b}}$ is $\frac{\pi}{6}.$

Determine P(E|F) in Exercises.
Two coins are tossed once, where
E : no tail appears, F : no head appears.

The mathematics department has 8 graduate assistants who are assigned to the same office. Each assistant is just as likely to study at home as in office. How many desks must there be in the office so that each assistant has a desk at least 90% of the time?

Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?

Evaluate the following integrals:$\int(\text{x}+1)\log\text{x dx}$