Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following intregals:
$\int\frac{\text{x}^2}{\text{x}^4-\text{x}^2-12}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{\text{x}^2}{\text{x}^4-\text{x}^2-12}\ \text{dx}$We express
$​​\frac{\text{x}^2}{\text{x}^4-\text{x}^2-12}=\frac{\text{x}^2}{\text{x}^4-4\text{x}^2+3\text{x}^2-12}$
$=\frac{\text{x}^2}{(\text{x}^2-4)(\text{x}^2+3)}$
$=\frac{\text{A}}{\text{x}^2-4}+\frac{\text{B}}{\text{x}^2+3}$
$\Rightarrow\text{x}^2=\text{A}(\text{x}^2+3)+\text{B}(\text{x}^2-4)$
Equating the coefficients of $x^2$​​​​​​​ and constant, we get
1 = A + B and 0 = 3A - 4B or
$\text{A}=\frac{4}{7}$ and $\text{B}=\frac{3}{7}$
$\therefore\text{I}=\int\bigg(\frac{\frac{4}{7}}{\text{x}^2-4}+\frac{\frac{3}{7}}{\text{x}^2+3}\bigg)\text{dx}$
$=\frac{4}{7}\int\frac{1}{\text{x}^2-4}\ \text{dx}+\frac{3}{7}\int\frac{1}{\text{x}^2+3}\text{dx}$
$=\frac{4}{7}\times\frac{1}{4}\log\Big|\frac{\text{x}-2}{\text{x}+2}\Big|+\frac{\sqrt{3}}{7}\tan^{-1}\frac{\text{x}}{\sqrt{3}}+\text{C}$
$=\frac{1}{7}\log\Big|\frac{\text{x}-2}{\text{x}+2}\Big|+\frac{\sqrt{3}}{7}\tan^{-1}\frac{\text{x}}{\sqrt{3}}+\text{C}$
Hence, $\int\frac{\text{x}^2}{\text{x}^4-\text{x}^2-12}\ \text{dx}=\frac{1}{7}\log\Big|\frac{\text{x}-2}{\text{x}+2}\Big|+\frac{\sqrt{3}}{7}\tan^{-1}\frac{\text{x}}{\sqrt{3}}+\text{C}$

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 "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

AB is a diameter of a circle and C is any point on the circle. Show that the area of $\triangle\text{ABC}$ is maximum, when it is isosceles.
→
The probability that a certain kind of component will survive a given shock test is $\frac{3}{4}.$ Find the probability that among 5 components tested.
  1. exactly 2 will survive.
  2. at most 3 will survive.
→
If $\text{xy}=1,$ prove that $\frac{\text{dy}}{\text{dx}}+\text{y}^2=0$
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = \sec(\text{x}+\text{y})$
→
Find the particular solution of the differential equation $(\text{x}-\text{y})\frac{\text{dy}}{\text{dx}}=\text{x +2y},$ given that when x = 1, y = 0.
→
If $a_1, a_2, a_3, ..., a_n$ is an arithmetic progression with common difference d, then evaluate the following expression.
$\tan\bigg[\tan^{-1}\Big(\frac{\text{d}}{1+\text{a}_1\text{a}_2}\Big)+\tan^{-1}\Big(\frac{\text{d}}{1+\text{a}_2\text{a}_3}\Big)+\tan^{-1}\Big(\frac{\text{d}}{1+\text{a}_3\text{a}_4}\Big)+\ .....\ +\tan^{-1}\Big(\frac{\text{d}}{1+\text{a}_{\text{n}-1}\text{a}_\text{n}}\Big)\bigg]$
→
Using matrices, solve the following system of equations:
2x + 3y + 3z = 5, x – 2y + z = – 4, 3x – y – 2z = 3.
→
Find the radius of the circular section of the sphere $|\vec{\text{r}}|=5$ by the plane $\vec{\text{r}}.(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=3\sqrt{3}.$
→
Evaluvate the following intregals:
$\int\frac{1}{\text{x}(\text{x}-2)(\text{x}-4)}\ \text{dx}$
→
Show that the following system of linear equation is inconsistent:
$2x + 5y = 7$
$6x + 15y = 13$
→