int frac{e^{sqrt{x}} cos(e^{sqrt{x}})}{sqrt{x | vedclass

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Question

(A) Let $I = \int \frac{e^{\sqrt{x}} \cos(e^{\sqrt{x}})}{\sqrt{x}} dx$. Substitute $t = e^{\sqrt{x}}$. Differentiating both sides with respect to $x$,

Vedclass · Accepted Answer

$2 \sin(e^{\sqrt{x}}) + C$

Vedclass · Answer

(A) Let $I = \int \frac{e^{\sqrt{x}} \cos(e^{\sqrt{x}})}{\sqrt{x}} dx$. Substitute $t = e^{\sqrt{x}}$. Differentiating both sides with respect to $x$,we get: $dt = e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}} dx$. This implies $\frac{e^{\sqrt{x}}}{\sqrt{x}} dx = 2 dt$. Substituting these into the integral: $I = \int \cos(t) \cdot 2 dt = 2 \int \cos(t) dt$. Integrating $\cos(t)$,we get $2 \sin(t) + C$. Substituting back $t = e^{\sqrt{x}}$,we get $2 \sin(e^{\sqrt{x}}) + C$.

$\int \frac{e^{\sqrt{x}} \cos(e^{\sqrt{x}})}{\sqrt{x}} dx = $

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