MCQ
$\int_0^a {\frac{{x\,dx}}{{\sqrt {{a^2} + {x^2}} }}} = $
- ✓$a\,(\sqrt 2 - 1)$
- B$a\,(1 - \sqrt 2 )$
- C$a\,(1 + \sqrt 2 )$
- D$2a\sqrt 3 $
$\Rightarrow 2xdx = dt,$ then
$\int_0^a {\frac{{xdx}}{{\sqrt {{a^2} + {x^2}} }} = \frac{1}{2}\int_{{a^2}}^{2{a^2}} {\frac{1}{{\sqrt t }}dt} } $
$ = [{(2{a^2})^{1/2}} - {a^{2/2}}] = a(\sqrt 2 - 1)$.
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$ \mathrm{L}_1: \overrightarrow{\mathrm{r}}=(2+\lambda) \hat{\mathrm{i}}+(1-3 \lambda) \hat{\mathrm{j}}+(3+4 \lambda) \hat{\mathrm{k}}, \lambda \in \mathbb{R} $
$ \mathrm{L}_2: \overrightarrow{\mathrm{r}}=2(1+\mu) \hat{\mathrm{i}}+3(1+\mu) \hat{\mathrm{j}}+(5+\mu) \hat{k}, \mu \in \mathbb{R}$
વચ્ચેનું ન્યૂનતમ અંતર $\frac{m}{\sqrt{n}}$ હોય, જ્યાં $\operatorname{gcd}(m, n)=1$, તો $m+n$ નું મૂલ્ય ........... છે.