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substitution integral of (e^x)/(e^x+e^{-x)}, u=e^x

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`x`²	`x`^□	log_□	√☐	^□√☐	≤	≥	□□	·	÷	`x`^◦	π
(☐)^′	`ddx`	∂∂`x`	∫	∫_□^□	lim	∑	∞	`θ`	(`f` ◦ `g`)	`f`(`x`)

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≥	≤	·	÷	`x`^◦	(☐)	\|☐\|	(`f` ◦ `g`)	`f`(`x`)	ln	`e`^☐
(☐)^′	∂∂`x`	∫_□^□	lim	∑	sin	cos	tan	cot	csc	sec

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substitution y^{′ ′}+3y^′=0

Solution

y=−13 e^c₁−3t+c₂

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Find vertex using polynomial form

Find vertex using parabola form

Find vertex using vertex form

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y=4x²+1

Parabola equation in polynomial form

The vertex of an up-down facing parabola of the form y=ax²+bx+c is x_v=−b2a

The parabola parameters are:

a=4, b=0, c=1

x_v=−b2a

x_v=−02· 4

بسّط

x_v=0

Plug in x_v=0 to find the y_v value

y_v=1

لذلك رأس القطع المكافئ هو

(0, 1)

إذًا الرأس هو نقطة عظمى a<0 إذا تحقّق أنّ
إذًا الرأس هو نقطة صغرى a>0 إذا تحقّق أنّ
a=4

صغرى (0, 1)

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−▭▭	<	7	8	9	÷	`AC`
+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

substitution integral of (e^x)/(e^x+e^{-x)}, u=e^x

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