MCQ
$\int_{ - \pi /2}^{\pi /2} {{{\sin }^2}x{{\cos }^2}x(\sin x + \cos x)\,dx = } $
- A$\frac{2}{{15}}$
- ✓$\frac{4}{{15}}$
- C$\frac{6}{{15}}$
- D$\frac{8}{{15}}$
$= \int_{ - \pi /2}^{\pi /2} {{{\sin }^3}x{{\cos }^2}xdx + \int_{ - \pi /2}^{\pi /2} {{{\sin }^2}x{{\cos }^3}x\,dx} } $
$ = 0 + 2\int_0^{\pi /2} {{{\sin }^2}x{{\cos }^3}xdx} $
$ = 0 + 2 \times \frac{2}{{15}} = \frac{4}{{15}}$ .
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$f(x)\, = \left\{ {\begin{array}{*{20}{c}}{x\,\sin \,\left( {\frac{1}{x}}\right)\,\,\,\,\,\,\,for\,\, - 1 \le x \le 1\,\,and\,\,x \ne \,0}\\
{0\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,0}
\end{array}} \right.$
$g(x)\, = \left\{ {\begin{array}{*{20}{c}}{{x^2}\,\sin \,\left( {\frac{1}{x}} \right)\,\,\,\,\,\,\,for\,\, - 1 \le x \le 1\,\,and\,\,x \ne \,0}\\{0\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,0}\end{array}} \right.$ $h (x) = | x |^3$ for $- 1 \le x \le 1$ Which of these functions are differentiable at $x = 0$ ?
Statement-$2$ : Two lines are skew lines if there exists no plane passing through them.