MCQ
Solution of $\frac{{dy}}{{dx}} = \frac{{x\log {x^2} + x}}{{\sin y + y\,\,\cos y}}$ is
- ✓$y\sin y = {x^2}\log x + c$
- B$y\sin y = {x^2} + c$
- C$y\sin y = {x^2} + \log x + c$
- D$y\sin y = x\log x + c$
Separating the variables and integrating
$\int {(\sin y + y\cos y)dy = \int {(x\log {x^2} + x)dx} } $
==> $ - \cos y + y\sin y + \cos y$
$ = \frac{{{x^2}}}{2}\log {x^2} - \int {\frac{{{x^2}}}{2}.\frac{1}{{{x^2}}}.2xdx + \int {x\,dx + c} } $
==> $y\sin y = \frac{{{x^2}}}{2}2\log x - \int {x\,dx + \int {xdx + c} } $
==> $y\sin y = {x^2}\log x + c$.
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$f(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\|x-1|, & x \geq 0\end{array} \text { and } g(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\1, & x \geq 0\end{array}\right. \text {. }\right.$
Then (gof) (x) is
$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$
where a,b,c $\in R$ and $[t]$ denotes greatest integer less than or equal to $t.$ Then, which of the following statements is true $?$