Download App

Definite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDefinite Integrals3 Marks
Question
Evaluate the following definite integrals:
$\int_{0}^\limits{1}\frac{1}{2\text{x}^2+\text{x}+1}\text{ dx}$
✓

Answer

We have,
$\int_{0}^\limits{1}\frac{1}{2\text{x}^2+\text{x}+1}\text{ dx}$
$=\frac{1}{2}\int_{0}^\limits{1}\frac{1\text{ dx}}{\big(\text{x}^2+\frac{1}{2}\text{x}+\frac{1}{2}\big)}$
$=\frac{1}{2}\int_{0}^\limits{1}\frac{\text{dx}}{\big(\text{x}+\frac{1}{4}\big)^2+\frac{1}{2}-\frac{1}{16}}$ $\Big[\text{Adding }\frac{1}{16}\&\text{ substracting }\frac{1}{16}\text{ in numerator}\Big]$
$=\frac{1}{2}\int_{0}^\limits{1}\frac{\text{dx}}{\big(\text{x}+\frac{1}{4}\big)^2+\frac{7}{16}}$
$=\frac{1}{2}\int_{0}^\limits{1}\frac{\text{dx}}{\big(\text{x}+\frac{1}{4}\big)^2+\big(\frac{\sqrt{7}}{4}\big)^2}$
$=\frac{1}{2}\cdot\frac{4}{\sqrt{7}}\Bigg[\tan^{-1}\Bigg(\frac{\text{x}+\frac{1}{4}}{\frac{\sqrt{7}}4{}}\Bigg)\Bigg]^1_0$
$=\frac{2}{\sqrt{7}}\bigg\{\tan^{-1}\frac{5}{\sqrt{7}}-\tan^{-1}\Big(\frac{1}{\sqrt{7}}\Big)\bigg\}$

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

Solve the Linear Programming Problem graphically:
Minimize Z = -3x + 4y subject to $x + 2y \leq 8, \ 3x + 2y \leq 12, \ x \geq 0, \ y \geq 0.$
→
For each of the differential equations in find the general solution:
y log y dx – x dy = 0
→
Examine the consistency of the system of equations:
5x - y + 4z = 5
2x - 3y + 5z = 2
5x - 2y + 6z = -1
→
Show that the three lines with direction cosines $\frac{12}{13},\ \frac{-3}{13},\ \frac{-4}{13},\ \frac{4}{13},\ \frac{12}{13},\ \frac{3}{13},\ \frac{3}{13},\ \frac{-4}{13},\ \frac{12}{13}$ are mutually perpendicular.
→
Solve the following equation for x:
$\cos^{-1}\Big(\frac{\text{x}^2-1}{\text{x}^2+1}\Big)+\frac{1}{2}\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)=\frac{2\pi}{3}$
→
Find the value of $\tan^{-1}\Big(\frac{\text{x}}{\text{y}}\Big)-\tan^{-1}\Big(\frac{\text{x-y}}{\text{x+y}}\Big)$
→
Evaluate the integral: $\sec ^{-1} \sqrt{x} d x$
→
Write a value of $\int\text{x}^2\sin\text{x}^3\text{ dx}$
→
Find differentiation the following function w.r.t. $x$.$
\log (\sqrt{x-1}+\sqrt{x+1})
$
→
Find the shortest distance between the lines

$\vec r = \left( {\hat i + 2\hat j + \hat k} \right) + \lambda \left( {\hat i - \hat j + \hat k} \right)$

$\vec r = \left( {2\hat i - \hat j - \hat k} \right) + \mu \left( {2\hat i + \hat j + 2\hat k} \right)$

→