If y-etan x- then cos2x- y2-where y2 - d2ydx2- y1 - dydx

Y=e^tan x ⇒ nbsp;y_1=^2x· e^tan x ⇒cos^2x · y_1=y Again differentiating w.r.t x, we get ⇒cos^2x · y_2 2cos xsin x · y_1=y_1 ⇒cos^2 x · y_2=y_1 ·sin 2x+y_1 (∵ ...

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Standard XII

Mathematics

Integration by Substitution

If y=etan x, ...

Question

If $y = e^{tan x},$ then $({cos}^{2} x) \cdot y_{2} =$
$(where y_{2} = \frac{d^{2} y}{d x^{2}}, y_{1} = \frac{d y}{d x})$

(1 - sin 2 x) y_{1}

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(- 1 - sin 2 x) y_{1}

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(1 + sin 2 x) y_{1}

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None of these

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Solution

The correct option is C $(1 + sin 2 x) y_{1}$
$y = e^{tan x} \Rightarrow y_{1} = {sec}^{2} x \cdot e^{tan x} \Rightarrow {cos}^{2} x \cdot y_{1} = y$
Again differentiating w.r.t $x$ , we get
$\Rightarrow {cos}^{2} x \cdot y_{2} - 2 cos x sin x \cdot y_{1} = y_{1} \Rightarrow {cos}^{2} x \cdot y_{2} = y_{1} \cdot sin 2 x + y_{1} (∵ 2 sin x cos x = sin 2 x) ∴ {cos}^{2} x \cdot y_{2} = (1 + sin 2 x) y_{1}$

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Integration by Substitution

Standard XII Mathematics

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If y=etanx, then (cos2x)⋅y2= (where y2=d2ydx2,y1=dydx)

The correct option is C (1+sin2x)y1y=etanx⇒y1=sec2x⋅etanx⇒cos2x⋅y1=y Again differentiating w.r.t x, we get ⇒cos2x⋅y2−2cosxsinx⋅y1=y1⇒cos2x⋅y2=y1⋅sin2x+y1(∵2sinxcosx=sin2x)∴cos2x⋅y2=(1+sin2x)y1

If $y = e^{tan x},$ then $({cos}^{2} x) \cdot y_{2} =$
$(where y_{2} = \frac{d^{2} y}{d x^{2}}, y_{1} = \frac{d y}{d x})$