Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals3 Marks
Question
Evaluate the following integrals:
$\int\frac{1}{(\text{x}-1)\sqrt{\text{x}^2+1}}\text{ dx}$
✓

Answer

We have
$\text{I}=\int\frac{1}{(\text{x}-1)\sqrt{\text{x}^2+1}}\text{ dx}$
Putting $\text{x}-1=\frac{1}{\text{t}}$
$\text{dx}=-\frac{1}{\text{t}^2}\text{ dt}$
$\therefore\ \text{I}=\int\frac{-\frac{1}{\text{t}^2}\text{ dt}}{\big(\frac{1}{\text{t}}\big)\sqrt{\big(1+\frac{1}{\text{t}}}\big)^2+1}$
$=\int\frac{-\frac{1}{\text{t}}\text{ dt}}{\sqrt{1+\frac{1}{\text{t}^2}+\frac{2}{\text{t}}+1}}$
$=\int\frac{-\frac{1}{\text{t}}\text{ dt}}{\sqrt{\frac{\text{t}^2+1+2\text{t}+\text{t}^2}{\text{t}}}}$
$=\int\frac{-\text{dt}}{\sqrt{2\text{t}^2+2\text{t}+1}}$
$=-\frac{1}{\sqrt{2}}\int\frac{\text{dt}}{\sqrt{\text{t}^2+\text{t}+\frac{1}{2}}}$
$=-\frac{1}{\sqrt{2}}\int\frac{\text{dt}}{\sqrt{\text{t}^2+\text{t}+\frac{1}{4}-\frac{1}{4}+\frac{1}{2}}}$
$=-\frac{1}{\sqrt{2}}\int\frac{\text{dt}}{\big(\text{t}+\frac{1}{2}\big)^2+\big(\frac{1}{2}\big)^2}$
$=-\frac{1}{\sqrt{2}}\log\begin{vmatrix}\text{t}+\frac{1}{2}+\sqrt{\Big(\text{t}+\frac{1}{2}\Big)^2+\frac{1}{4}}\end{vmatrix}+\text{C}$ where $\text{t}=\frac{1}{\text{x}-1}$

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.(3 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

Evaluate the following integrals:
$\int\frac{1}{\text{x}}(\log\text{x})^2\text{dx}$
→
Evaluate the following integrals:
$\int\text{x}^2\text{e}^{\text{x}^3}\cos\big(\text{e}^{\text{x}^3}\big)\text{dx}$
→
Reduce the equation $\bar{r} \cdot(\hat{i}+8 \hat{j}+24 \hat{k})=13$ to normal form and hence find

(i) the length of the perpendicular from the origin to the plane (ii) direction cosines of the normal.

→
Prove the following results:
$\frac{9\pi}{4}-\frac{9}{4}\sin^{-1}\frac{1}{3}=\frac{9}{4}\sin^{-1}\frac{2\sqrt2}{3}$
→
Evaluate the following integrals:
$\int\frac{5\text{x}^4+12\text{x}^3+7\text{x}^2}{\text{x}^2+\text{x}}\text{dx}$
→
Integrate the following functions w.r.t. x:

$e^{\sin ^{-1} x}\left[\frac{x+\sqrt{1-x^2}}{\sqrt{1-x^2}}\right]$

→
Find the length of the perpendicular drow from the point (5, 4, -1) to the line $\vec{\text{r}}=\hat{\text{i}}+\lambda\big(2\hat{\text{i}}+9\hat{\text{j}}+5\hat{\text{k}}\big).$
→
$\int \sqrt{\frac{9+x}{9-x}} d x$
→
Form the differential equation of:all parabolas which have 4b as latus rectum and whose axis is parallel to Y-axis.
→
If $|\vec{\text{a}}|=2,\big|\vec{\text{b}}\big|=5$ and $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=8,$ find $\vec{\text{a}}.\vec{\text{b}}.$
→