Download App

Indefinite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\frac{(\text{x}^2+1)(\text{x}^2+4)}{(\text{x}^2+3)(\text{x}^2-5)}\ \text{dx}$
✓

Answer

$\text{I}=\int\frac{(\text{x}^2+1)(\text{x}^2+4)}{(\text{x}^2+3)(\text{x}^2-5)}\ \text{dx}$
Since,
$\frac{(\text{x}^2+1)(\text{x}^2+4)}{(\text{x}^2+3)(\text{x}^2-5)}=\frac{[(\text{x}^2+3)-][(\text{x}^2-5)+9]}{(\text{x}^2+3)(\text{x}^2-5)}$
$\Rightarrow\frac{(\text{x}^2+1)(\text{x}^2+4)}{(\text{x}^2+3)(\text{x}^2-5)}=\frac{(\text{x}^2+3)(\text{x}^2-5)+9(\text{x}^2+3)-2(\text{x}^2-5)-18}{(\text{x}^2+3)(\text{x}^2-5)}$
$\Rightarrow\frac{(\text{x}^2+1)(\text{x}^2+4)}{(\text{x}^2+3)(\text{x}^2-5)}=1+\frac{9}{(\text{x}^2-5)}-\frac{2}{(\text{x}^2+3)}-\frac{18}{(\text{x}^2+3)(\text{x}^2-5)}\ ...(1)$
Let $\text{I}_1=\int\frac{1}{(\text{x}^2+3)(\text{x}^2-5)}$ and $\text{x}^2=\text{y}$
$\Rightarrow\frac{1}{(\text{y}+3)(\text{y}-5)}=\frac{\text{A}}{(\text{y}+3)}+\frac{\text{B}}{(\text{y}-5 )}$
$=\frac{\text{A}(\text{y}-5)+\text{B}(\text{y}+3)}{(\text{y}+3)(\text{y}-5)}$
$\Rightarrow\frac{1}{(\text{y}+3)(\text{y}-5)}=\frac{(\text{A}+\text{B})\text{y}-(5\text{A}+3\text{B})}{(\text{y}+3)(\text{y}-5)}$
Compairing the coefficient, we get
A + B = 0 and 5A + 3B = -1
By solving the equations, we get
$\text{A}=-\frac{1}{8}$ and $\text{B}=\frac{1}{8}$
From (1) we get
$\text{I}=\int\Big[1+\frac{9}{(\text{x}^2-5)}-\frac{2}{(\text{x}^2+3)}-18\Big(\frac{-1}{8(\text{x}^2+3)}+\frac{1}{8(\text{x}^2-5)}\Big)\Big]\text{dx}$
$\Rightarrow\text{I}\int\Big[1+\frac{27}{4(\text{x}^2-5)}+\frac{1}{(\text{x}^2+3)}\Big]\text{dx}$
$\therefore\text{I}=\text{x}+\frac{27}{8\sqrt{5}}\ln\Big(\big|\frac{\text{x}-\sqrt{5}}{\text{x}+\sqrt{5}}\big|\Big)+\frac{1}{4\sqrt{3}}\tan^{-1}\Big(\frac{\text{x}}{\sqrt{3}}\Big)+\text{C}$

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 "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

If $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda,\mu$ so that $\text{A}^2=\lambda\text{A}+\mu\text{I}$
→
If $\vec{a}, \vec{b}, \vec{\mathrm{c}}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{c} \cdot \vec{d}=15$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec c$.
→
Find $\frac{d y}{d x}$
$(a) \ y=\sin x^{\sin x^{\sin x ...... \infty}}$
$(b) \ y=\sqrt{\log _e x+\sqrt{\log _e x+\sqrt{\log _e x+\ldots \ldots \ldots \infty}}}$
$(c)\ y=e^{x+e^{x+e^{x+\ldots \infty}}}$
→
Evaluate the following integrals:
$\int^\limits2_1\frac{1}{\text{x}(1+\log\text{x})^2}\text{ dx}$
→
Integrate the function in Exercise:
$\frac{1}{\text{x}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}}$
$\Bigg[\text{Hint:}\frac{1}{\text{x}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}}=\frac{1}{\text{x}^{\frac{1}{3}}\bigg(1+\text{x}^{\frac{1}{6}}\bigg)},\text{put}\ \text{x}=\text{t}^{6}\Bigg]$
→
Find the points at which the function $f$ given by $f(x) = (x - 2)^4 (x + 1)^3$ has
  1. local maxima
  2. local minima
  3. point of inflexion
→
Evaluate the following integrals:
$\int_{0}^\limits{1}\text{x}\log(1+2\text{x})\text{dx}$
→
If $\text{x}=\text{a}\sin\text{t}\ \text{and}\ \text{y}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2}),$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$
→
Show that the differential equation of $\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y d x=\left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x d y$ is homogeneous and solve it.
→
Using differentials, find the approximate values of the following:
$\log_\text{e}10.01$ it being given that $\log_{10}=0.4343$
→