Evaluate the following integrals: ^ 4 _2 x x^2+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{4}_2\frac{\text{x}}{\text{x}^2+1}\text{dx}$

✓

Answer

$\int\limits^{4}_2\frac{\text{x}}{\text{x}^2+1}\text{dx}$
$=\frac{1}{2}\int\limits^{4}_2\frac{2\text{x}}{\text{x}^2+1}\text{dx}$
$=\frac{1}{2}\times\Big[\log(\text{x})^2-1\Big]^4_2$ $\Big[\int\frac{\text{f}'(\text{x})}{\text{f(x)}}\text{ dx}=\log\text{f(x)}+\text{C}\Big]$
$=\frac{1}{2}\big(\log17-\log5\big)$
$=\frac{1}{2}\log\Big(\frac{17}{5}\Big)$ $\Big(\log\text{a}-\log\text{b}=\log\frac{\text{a}}{\text{b}}\Big)$

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 "2 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

Three numbers are chosen from 1 to 20. Find the probability that they are consecutive.

Verify Rolle’s theorem for the function $\text{f(x)}=\text{x}^2+2\text{x}-8,\text{x}\in[-4,2].$

Prove that the function defined by f(x) = tan x is a continuous function.

Evaluate $\int\frac{\text{x}^2}{1+\text{x}^3}\text{ dx}$

Show that the lines $\frac{{x - 5}}{7} = \frac{{y + 2}}{{ - 5}} = \frac{z}{1}$ and $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ are perpendicular to each other.

If a line makes angles 90° and 60° respectively with the positive direction of x and y axes, find the angle which it makes with the positive direction of z-axis.

Write the value of $\tan^{-1}\Big\{2\sin\Big(2\cos^{-1}\frac{\sqrt3}{2}\Big)\Big\}$

Write the value of $\hat{\text{i}}.\big(\hat{\text{j}}\times\hat{\text{k}}\big)+\hat{\text{j}}.\big(\hat{\text{k}}\times\hat{\text{i}}\big)+\hat{\text{k}}.\big(\hat{\text{i}}\times\hat{\text{j}}\big).$

Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.

Write the angle between the vectors $\vec{\text{a}}\times\vec{\text{b}}$ and $\vec{\text{b}}\times\vec{\text{a}}.$