The value of int {frac{{sqrt {{x^2} - {a^2}} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $I = \int {\frac{{\sqrt {{x^2} - {a^2}} }}{x}dx}$.Substitute $\sqrt {{x^2} - {a^2}} = t$,then ${x^2} - {a^2} = {t^2}$,which implies ${x^2} = {

Vedclass · Accepted Answer

$\sqrt {{x^2} - {a^2}} - a{	an ^{ - 1}}\left( {\frac{{\sqrt {{x^2} - {a^2}} }}{a}} ight) + C$

Vedclass · Answer

(A) Let $I = \int {\frac{{\sqrt {{x^2} - {a^2}} }}{x}dx}$.Substitute $\sqrt {{x^2} - {a^2}} = t$,then ${x^2} - {a^2} = {t^2}$,which implies ${x^2} = {a^2} + {t^2}$.Differentiating both sides,$2xdx = 2tdt$,so $xdx = tdt$.Now,multiply the numerator and denominator by $x$:$I = \int {\frac{{\sqrt {{x^2} - {a^2}} \cdot x}}{{{x^2}}}dx} = \int {\frac{t}{{{a^2} + {t^2}}} \cdot tdt} = \int {\frac{{{t^2}}}{{{a^2} + {t^2}}}dt}$.$I = \int {\left( {\frac{{{t^2} + {a^2} - {a^2}}}{{{a^2} + {t^2}}}} ight)dt} = \int {\left( {1 - \frac{{{a^2}}}{{{a^2} + {t^2}}}} ight)dt}$.$I = \int {1dt} - {a^2}\int {\frac{1}{{{a^2} + {t^2}}}dt} = t - {a^2} \cdot \frac{1}{a}{	an ^{ - 1}}\left( {\frac{t}{a}} ight) + C$.$I = t - a{	an ^{ - 1}}\left( {\frac{t}{a}} ight) + C$.Substituting back $t = \sqrt {{x^2} - {a^2}}$,we get:$I = \sqrt {{x^2} - {a^2}} - a{	an ^{ - 1}}\left( {\frac{{\sqrt {{x^2} - {a^2}} }}{a}} ight) + C$.

The value of $\int {\frac{{\sqrt {{x^2} - {a^2}} }}{x}dx} $ is:

Similar Questions

$\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x=$

If $\int \frac{\sqrt{1-x^2}}{x^4} \,dx = A(x)\left(\sqrt{1-x^2}\right)^{m} + c$ for a suitably chosen integer $m$ and a function $A(x)$,where $c$ is a constant of integration,then $(A(x))^{m}$ equals

$\int \frac{x}{\sqrt{1-2 x^4}} \, dx = $ (Where $C$ is a constant of integration)

If $\int \frac{1}{x\left[(\log x)^2+4 \log x-1\right]} d x=A \log \left[\frac{\log x+B}{\log x+C}\right]+K$ where $K$ is the constant of integration,then

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module