Download App

Indefinite Integrals — Maths STD 12 Science — Question

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

Answer

Consider the integral
$\text{I}=\int\frac{2\text{x}^2+1}{\text{x}^2(\text{x}^2+4)}\ \text{dx}$
Now let us seoarate the fraction $\frac{2\text{x}^2+1}{\text{x}^2(\text{x}^2+4)}$ through partial fractions.
Substitute $x^2 = t$, then
$\frac{2\text{x}^2+1}{\text{x}^2(\text{x}^2+4)}=\frac{2\text{t}+1}{\text{t}(\text{t}+4)}$
$\Rightarrow\frac{2\text{t}+1}{\text{t}(\text{t}+4)}=\frac{\text{A}}{\text{t}}+\frac{\text{B}}{\text{t}+4}$
$\Rightarrow\frac{2\text{t}+1}{\text{t}(\text{t}+4)}=\frac{\text{A}(\text{t}+4)+\text{Bt}}{\text{t}(\text{t}+4)}$
$=2\text{t}+1=\text{A}(\text{t}+4)+\text{Bt}$
$=2\text{t}+1=\text{At}+4\text{A}+\text{Bt}$
Compairing the coefficient, we have,
$A + B = 2 $and $4A = 1$
$\Rightarrow\text{A}=\frac{1}{2}$ and $\text{B}=\frac{7}{4}$
$\Rightarrow\frac{2\text{x}^2+1}{\text{x}^2(\text{x}^2+4)}=\frac{1}{4\text{x}^2}+\frac{7}{4(\text{x}^2+4)}$
Thus, we have,
$\text{I}=\int\frac{2\text{x}^2+1}{(\text{x}^2+4)}\ \text{dx}$
$=\frac{1}{4}\int\frac{\text{dx}}{\text{x}^2}+\frac{7}{4}\int\frac{\text{dx}}{(\text{x}^2+4)}$
$=-\frac{1}{4\text{x}}+\frac{7}{4}\times\frac{1}{2}\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}$
$=-\frac{1}{4\text{x}}+\frac{7}{8}\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}$

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

If $\text{x}=\cot\text{t and y}=\sin\text{t},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{3}}\text{ at t}=\frac{2\pi}{3}$
→
Prove that:
$\begin{vmatrix}\text{b}+\text{c}&\text{a}-\text{b}&\text{a}\\\text{c}+\text{a}&\text{b}-\text{c}&\text{b}\\\text{a}+\text{b}&\text{c}-\text{a}&\text{c}\end{vmatrix}=3\text{abc}-\text{a}^3-\text{b}^3-\text{c}^3$
→
If $\text{A}=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix},$ find $A^2.$
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\frac{1+\text{y}^2}{\text{y}^3}$
→
Evaluate the following integrals:$\int\limits^2_{-2}\frac{3\text{x}^3+2|\text{x}|+1}{\text{x}^2+|\text{x}|+1}\text{ dx}$
→
If three numbers are added, their sum is $2$. If $2$ times the second number is subtracted from the sum of first and third numbers, we get $8$ and if three times the first number is added to the sum of second and third numbers, we get $4$. Find the numbers using matrices.
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}\text{ on }[0,\pi]$
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\sin^2\text{x}=\frac{1}{\text{x}\log\text{x}}$
→
Prove the following :

$\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{3}\right)=\frac{\pi}{4}$

→
If $\vec{\text{p}}$ and $\vec{\text{q}}$ are unit vectors forming an angle of 30°; find the area of the parallelogram having $\vec{\text{a}}=\vec{\text{p}}+2\vec{\text{q}}$ and $\vec{\text{b}}=2\vec{\text{p}}+\vec{\text{q}}$ as its diagonals.
→