38. Match the following\\
(i) $\int _ { 0 } ^ { \pi / 2 } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \left( \cos \mathrm { x } \cot \mathrm { x } - \log ( \sin \mathrm { x } ) ^ { \sin \mathrm { x } } \right) \mathrm { dx }$\\
(A) 1\\
(ii) Area bounded by $- 4 y ^ { 2 } = x$ and $x - 1 = - 5 y ^ { 2 }$\\
(B) 0\\
(iii) Cosine of the angle of intersection of curves $y = 3 ^ { x - 1 } \log x$ and

$$y = x ^ { x } - 1 \text { is }$$

(C) $6 \ln 2$\\
(iv) Data could not be retrieved.\\
(D) $4 / 3$

Sol. (i) $I = \int _ { 0 } ^ { \pi / 2 } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \left( \cos \mathrm { x } \cdot \cot \mathrm { x } - \log ( \sin \mathrm { x } ) ^ { \sin \mathrm { x } } \right) \mathrm { dx }$

$$\Rightarrow \quad \mathrm { I } = \int _ { 0 } ^ { \pi / 2 } \frac { \mathrm {~d} } { \mathrm { dx } } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \mathrm { dx } = 1 .$$

(ii) The points of intersection of $- 4 y ^ { 2 } = x$ and $x - 1 = - 5 y ^ { 2 }$ is $( - 4 , - 1 )$ and $( - 4,1 )$

Hence required area $= 2 \left[ \mid \int _ { 0 } ^ { 1 } \left( 1 - 5 y ^ { 2 } \right) d y - \int _ { 0 } ^ { 1 } - 4 y ^ { 2 } d y \right] \left\lvert \, = \frac { 4 } { 3 } \right.$.\\
(iii) The point of intersection of $y = 3 ^ { x - 1 } \log x$ and $y = x ^ { x } - 1$ is $( 1,0 )$

Hence $\frac { d y } { d x } = \frac { 3 ^ { x - 1 } } { x } + 3 ^ { x - 1 } \log 3 \cdot \log x . \left. \quad \frac { d y } { d x } \right| _ { ( 1,0 ) } = 1$\\
for $\mathrm { y } = \mathrm { x } ^ { \mathrm { x } } - \left. 1 \cdot \frac { \mathrm { dy } } { \mathrm { dx } } \right| _ { ( 1,0 ) } = 1$\\
If $\theta$ is the angle between the curve then $\tan \theta = 0 \Rightarrow \cos \theta = 1$.\\
(iv) $\frac { \mathrm { dy } } { \mathrm { dx } } = \left( \frac { 2 } { \mathrm { x } + \mathrm { y } } \right)$\\
$\Rightarrow \frac { \mathrm { dx } } { \mathrm { dy } } - \frac { \mathrm { x } } { 2 } = \frac { \mathrm { y } } { 2 }$\\
$\Rightarrow \quad \mathrm { xe } ^ { - \mathrm { y } / 2 } = \frac { 1 } { 2 } \int \mathrm { y } \cdot \mathrm { e } ^ { - \mathrm { y } / 2 } \mathrm { dy }$\\
$\Rightarrow \quad x + y + 2 = k ^ { \mathrm { y } / 2 } = 3 \mathrm { e } ^ { \mathrm { y } / 2 }$.\\