Question

guys plez answer this question......​

Answer 1

Answer:

The natural environment encompasses all living and non-living things occurring naturally, meaning in this case not artificial. The term is most often applied to the Earth or some parts of Earth.

Answer 2

$\huge{\boxed {\mathbb {QUESTION}}}$

$Find\:an\: anti \:derivative\:(or \:integral)\:of the \\following\: functions\: by \:the\: method\: of\: inspection$

$1)\:sin\:2x$

$2)\:cos\:3x$

$3)\:{e}^{2x}$

$4)\:{(ax+b)}^{2}$

$5)\:sin\:2x-4\:{e}^{3x}$

$\huge{\boxed {\mathbb {ANSWER}}}$

$\huge {1)\:sin2x}$

$The\: anti\: derivative\: of\: sin\: 2x\: is\: a\: function\: of\: x$

$\implies \frac{d}{dx} (cos\:2x)=-2sin\:2x$

$\implies sin\:2x=-\frac{1}{2}\frac{d}{dx} (cos\:2x)$

$\implies {sin\:2x=\frac{d}{dx} (-\frac{1}{2} cos\:2x)}$

$\therefore Anti\:derivative\:of\:sin\:2x\:is{\boxed {\boxed {(-\frac{1}{2} cos\:2x)}}}$

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$\huge{ 2)\:cos\:3x}$

$The\: anti\: derivative\: of\: cos\: 3x\: is\: a\: function\: of\: x$

$\implies \frac{d}{dx} (sin\:3x)=3cos\:3x$

$\implies cos\:3x=\frac{1}{3}\frac{d}{dx} (sin\:3x)$

$\implies cos\:3x=\frac{d}{dx}(\frac{1}{3} (sin\:3x)$

$\therefore Anti\:derivative\:of\:cos\:3x\:is{\boxed {\boxed{(\frac{1}{3} (sin\:3x)}}}$

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$\huge {3)\:{e}^{2x}}$

$The\: anti\: derivative\: of\:{e}^{2x} \: is\: a\: function\: of\: x$

$\implies \frac{d}{dx} {e}^{2x}=2{e}^{2x}$

$\implies {e}^{2x}=\frac{1}{2}\frac{d}{dx}({e}^{2x})$

$\implies {e}^{2x}=\frac{d}{dx}(\frac{1}{2}{e}^{2x})$

$\therefore Anti\:derivative\:of\:{e}^{2x}\:is{\boxed {\boxed{(\frac{1}{2}{e}^{2x})}}}$

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$\huge {4)\:{(ax+b)}^{2}}$

$The\: anti\: derivative\: of\:{(ax+b)}^{2} \: is\: a\: function\: of\: x$

$\implies \frac{d}{dx}{(ax+b)}^{2}=3a{(ax+b)}^{2}$

$\implies {(ax+b)}^{2}=\frac{1}{3a}\frac{d} {dx} {(ax+b)}^{3}$

$\implies {(ax+b)}^{2}=\frac{d} {dx}(\frac{1}{3a} {(ax+b)}^{3})$

$\therefore Anti\:derivative\:of\:{(ax+b)}^{2}\:is{\boxed {\boxed{(\frac{1}{3a} {(ax+b)}^{3})}}}$

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$\huge{5)\:sin\:2x-4\:{e}^{3x}}$

$The\: anti\: derivative\: of\:\: is\:sin\:2x-4\:{e}^{3x} a\: function\: of\: x$

$\implies \frac{d}{dx} (-\frac{1}{2}cos\:2x-\frac{4}{3}{e}^{3x}) =sin\:2x-4{e}^{3x}$

$\therefore Anti\:derivative\:of\:sin\:2x-4\:{e}^{3x}\:is{\boxed {\boxed{(-\frac{1}{2}cos\:2x-\frac{4}{3}{e}^{3x})}}}$

$\huge{\boxed {\mathbb {HOPE \:IT \:HELPS \:YOU }}}$

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$\huge{\boxed {\mathbb {EXTRA\:INFORMATION}}}$

$Inspection\: method$

• $An\: equation \:consisting \:of \:two\: sides\\ (LHS \:and \:RHS) that \:are\\\:equal\: to \:each \:other.$

• $When\: solving\: an \:equation\: we\\ try\: to\: find \:the \:number\: that\:when\\ substituted\:will\: make \:a \:true \:sentence.$

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