Download App

Indefinite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIndefinite Integrals5 Marks
Question
Evaluate the following integrals: $\int\frac{1}{\text{x}^4+\text{x}^2+1}\ \text{dx}$
✓

Answer

We have
$\text{I}=\int\frac{1}{\text{x}^4+\text{x}^2+1}\ \text{dx}$
$=\frac{1}{2}\int\frac{2\text{dx}}{\text{x}^4+\text{x}^2+1}$
$\Rightarrow\int\bigg(\frac{(\text{x}^2+1)-(\text{x}^2-1)}{\text{x}^4+\text{x}^2+1}\bigg)\text{dx}$
$\Rightarrow\frac{1}{2}\int\Big(\frac{\text{x}^2+1}{\text{x}^2+\text{x}^2+1}\Big)\text{dx}-\frac{1}{2}\int\Big(\frac{\text{x}^2-1}{\text{x}^4+\text{x}^2+1}\Big)\text{dx}$
$\Rightarrow\frac{1}{2}\int\Big(\frac{\text{x}^2+1}{\text{x}^4+\text{x}^2+1}\Big)\text{dx}-\frac{1}{2}\int\Big(\frac{\text{x}^2-1}{\text{x}^4+\text{x}^2+1}\Big)\text{dx}$
Dividing numerator and denominator bt $x^2$
$\text{I}=\frac{1}{2}\int\Bigg(\frac{1+\frac{1}{\text{x}^2}}{\text{x}^2+1+\frac{1}{\text{x}^2}}\Bigg)\text{dx}-\frac{1}{2}\int\Bigg(\frac{1-\frac{1}{\text{x}^2}}{\text{x}^2+1+\frac{1}{\text{x}^2}}\Bigg)\text{dx}$
$=\frac{1}{2}\int\Bigg(\frac{1+\frac{1}{\text{x}^2}}{\text{x}^2+\frac{1}{\text{x}^2}-2+3}\Bigg)\text{dx}-\frac{1}{2}\int\frac{\Big(1-\frac{1}{\text{x}^2}\Big)\text{dx}}{\text{x}^2+\frac{1}{\text{x}^2}+2-1}$
$=\frac{1}{2}\int\frac{\Big(1+\frac{1}{\text{x}^2}\Big)\text{dx}}{\Big(\text{x}-\frac{1}{\text{x}}\Big)^2+(\sqrt{3})^2}-\frac{1}{2}\int\frac{\Big(1-\frac{1}{\text{x}^2}\Big)\text{dx}}{\Big(\text{x}+\frac{1}{\text{x}}\Big)-1^2}$
putting $\text{x}-\frac{1}{\text{x}}=\text{t}$
$\Rightarrow\Big(1-\frac{1}{\text{x}^2}\Big)\text{dx}=\text{dp}$
$\therefore\text{I}=\frac{1}{2}\int\frac{\text{dt}}{\text{t}^2+(\sqrt{3})^2}-\frac{1}{2}\int\frac{\text{dp}}{\text{p}^2-1^2}$
$=\frac{1}{2}\times\frac{1}{\sqrt{3}}\tan^{-1}\Big(\frac{\text{t}}{\sqrt{3}}\Big)-\frac{1}{2}\times\frac{1}{2\times1}\Big|\frac{\text{p}-1}{\text{p}+1}\Big|+\text{C}$
$=\frac{1}{2\sqrt{3}}\tan^{-1}\Big(\frac{\text{x}-\frac{1}{\text{x}}}{\sqrt{3}}\Big)-\frac{1}{4}\log\Bigg|\frac{\text{x}+\frac{1}{\text{x}}-1}{\text{x}+\frac{1}{\text{x}}+1}\Bigg|+\text{C}$
$=\frac{1}{2\sqrt{3}}\tan^{-1}\Big(\frac{\text{x}^2-1}{\text{x}\sqrt{3}}\Big)-\frac{1}{4}\log\Big|\frac{\text{x}^2-\text{x}+1}{\text{x}^2+\text{x}+1}\Big|+\text{C}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b - 4. Prove that * is neither commutative nor associative on Z.
→
Find an equation for the set all points that are equidistant from the planes $3x - 4y + 12z = 6$ and $4x + 3z = 7$
→
Prove that the semi $-$ vertical angle of the right circular cone of given volume and least curved surface area is $\cot ^{-1} \sqrt{2}$
→
$A$ and $B$ take turns in throwing two dice, the first to throw $10$ being awarded the prize, show that if $A$ has the first throw, their chance of winning are in the ratio $12 : 11$.
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{\text{x}}{2}-\sin\frac{\pi\text{x}}{6}\text{ on }[-1,0]$
→
Prove that: $\begin{vmatrix}\text{a}&\text{a}+\text{b}&\text{a}+2\text{b} \\\text{a}+2\text{b}&\text{a}&\text{a}+\text{b}\\\text{a}+\text{b}&\text{a}+2\text{b}&\text{a} \end{vmatrix}=9(\text{a}+\text{b})\text{b}^2$
→
If $\text{A}=\begin{bmatrix}3 & 1 \\-1 & 2 \end{bmatrix},$ Show that $A^2 - 5A + 7I = 0.$ Hence, find $A^{-1}.$
→
Differentiate $\tan^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big)$ with respect to $\sqrt{1-\text{x}^2},$ if -1 < x < 1.
→
Find which of the function:
$\text{f(x)}=\begin{cases}\frac{2\text{x}^2-3\text{x}-2}{\text{x}-2},&\text{if x}\neq2\\5,&\text{if x}=2\end{cases}$
at x = 2
→
Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b - ab, for all a, b ∈ S.
Prove that:
  1. * is a binary operation on S.
  2. * is commutative as well as associative.
→