Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\frac{(\text{x}-1)^2}{\text{x}^4+\text{x}^2+1}\text{ dx}$
✓

Answer

Let $\text{I}=\int\frac{(\text{x}-1)^2}{\text{x}^4+\text{x}^2+1}\text{ dx}$
$=\frac{\text{x}^2-2\text{x}+1}{\text{x}^4+1+\frac{1}{\text{x}^2}}\text{ dx}$
Dividing numerator and denominator by $x^2$​​​​​​​
$\therefore\text{I}=\int\frac{1-\frac{2}{\text{x}}+\frac{1}{\text{x}^2}}{\text{x}^2+1+\frac{1}{\text{x}^2}}\ \text{dx}$
$=\int\frac{\Big(1+\frac{1}{\text{x}^2}\Big)}{\Big(\text{x}-\frac{1}{\text{x}}\Big)^2+3}\ \text{dx}-\int\frac{2\text{x}}{\text{x}^4+\text{x}^2+1}\ \text{dx}$
Let $\Big(\text{x}-\frac{1}{\text{x}}\Big)=\text{t}$
$\Rightarrow\Big(1+\frac{1}{\text{x}^2}\Big)\text{dx}=\text{dt}$ [for 1st part]
Let $\text{x}^2=\text{z}$
$\Rightarrow2\text{x dx}=\text{dz}$ [For 2nd part]
$\therefore\text{I}=\int\frac{\text{dt}}{\text{t}^2+3}-\int\frac{\text{dz}}{\text{z}^2+\text{z}+1}$
$=\int\frac{\text{dt}}{\text{t}^3+3}-\int\frac{\text{dz}}{\Big(\text{z}+\frac{1}{2}\Big)^2+\frac{3}{4}}$
$\Rightarrow\text{I}=\frac{1}{\sqrt{3}}\tan^{-1}\Big(\frac{\text{t}}{\sqrt{3}}\Big)-\frac{2}{\sqrt{3}}\tan^{-1}\Bigg(\frac{\text{z}+\frac{1}{2}}{\frac{\sqrt{3}}{2}}\Bigg)+\text{C}$
$\Rightarrow\text{I}=\frac{1}{\sqrt{3}}\tan^{-1}\Big(\frac{\text{x}^2-1}{\sqrt{3}\text{x}}\Big)-\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{2\text{z}+1}{\sqrt{3}}\Big)+\text{C}$
Hence,
$\text{I}=\frac{1}{\sqrt{3}}\tan^{-1}\Big(\frac{\text{x}^2-1}{\sqrt{3}\text{x}}\Big)-\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{2\text{x}^2+1}{\sqrt{3}}\Big)+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university.
  1. none will graduate.
  2. only one will graduate.
  3. all will graduate.
→
Find the second order derivatives of the following functions:
$\text{e}^{6\text{x}} \cos \text{x}$
→
Evaluate the following integrals:
$\int\frac{\text{x}^2+1}{\text{x}^4+\text{x}^2+1}\ \text{dx}$
→
Show that $\text{A}=\begin{bmatrix} 6 & 5 \\ 7 & 6 \end{bmatrix}$ satisfies the equation $x^2 - 12x + 1 = 0$. Thus, find $A^{-1}.$
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{2\text{a}^{\text{x}}}{1-\text{a}^{2\text{x}}}\Big),\text{a}>1, -\infty<\text{x}<0$
→
If $\text{x}=\text{e}^{\cos2\text{t}}$ and $\text{y}=\text{e}^{\sin2\text{t}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}\log\text{x}}{\text{x}\log\text{y}}$
→
Let A = {a, b, c}, B = {u, v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as:
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.
→
Find the equations of tangents and normals to the curve at the point on it.

$x \sin 2 y=y \cos 2 x$ at $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

→
A coin is tossed three times. Let the events A, B and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row. Check the independence of,
  1. A and B.
  2. B and C.
  3. C and A.
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=\text{x}^2\cot\text{x}+2\text{x}$
→