Evaluate the following integrals: _ 2 ^4 x x^2+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int_{2}^\limits{4}\frac{\text{x}}{\text{x}^2+1}\text{ dx}$

✓

Answer

Let $\text{x}^2=\text{t}$ Then, $2\text{x dx}=\text{dt}$
When $\text{x}=2,\text{ t}=4$ and $\text{x}=4,\text{ t}=16$
$\therefore\ \text{I}=\int_{2}^\limits{4}\frac{\text{x}}{\text{x}^2+1}\text{ dx}$
$\Rightarrow\text{I}=\int_{4}^\limits{16}\frac{1}{2}\frac{\text{dt}}{\text{t}+1}$
$\Rightarrow\text{I}=\frac{1}{2}\Big[\log(\text{t}+1)\Big]^{16}_4$
$\Rightarrow\text{I}=\frac{1}{2}\log17-\frac{1}{2}\log5$
$\Rightarrow\text{I}=\frac{1}{2}\log\frac{17}{5}$

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" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the following functions with respect to x:
$\log\Big\{\cot\Big(\frac{\pi}{4}+\frac{\pi}{2}\Big)\Big\}$

Find the value of x if the area of $\triangle$ is 35 square cms with vertices (x, 4), (2, -6) and (5, 4).

Each of the following defines a relation on N:
x, y is square of an integer $\text{x},\text{y}\in\text{N}$
Determine which of the above relations are reflexive, symmetric and transitive.

For what value of k is the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin5\text{x}}{3\text{x}}, &\text{if}\text{ x}\neq0\\\text{k}, &\text{if}\text{ x}=0\end{cases}$ is continuous at x = 0?

In a school there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in class XII given that the chosen student is a girl?

Find the integrals of the functions in Exercises:
$\cos2\text{x}\cos4\text{x}\cos6\text{x}$

A bag contains 6 red and 8 black balls and another bag contains 8 red and 6 black balls. A ball is drawn from the first bag and without noticing its colour is put in the second bag. A ball is drawn from the second bag. Find the probability that the ball drawn is red in colour.

A coin is tossed three times. Find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$ in each of the following:
A = At most two tails,
B = At least one tail.

Evaluate the following integrals:
$\int\sin^3\text{x}\cos^5\text{x}\text{ dx}$

Find the particular solution of the differential equation $\frac { d y } { d x }$+ 2y tan x = sin x, given that y = 0 when x = $\frac{\pi}{3}$.