int frac{e^{x}}{sqrt{x}}(1+2 x) d x= | vedclass

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(B) Let $I = \int \frac{e^{x}}{\sqrt{x}}(1+2x) dx$. We can rewrite the integrand as: $I = \int e^{x} \left( \frac{1}{\sqrt{x}} + \frac{2x}{\sqrt{x}} \

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$2 \sqrt{x} e^{x}+c$

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(B) Let $I = \int \frac{e^{x}}{\sqrt{x}}(1+2x) dx$. We can rewrite the integrand as: $I = \int e^{x} \left( \frac{1}{\sqrt{x}} + \frac{2x}{\sqrt{x}} \right) dx = \int e^{x} \left( \frac{1}{\sqrt{x}} + 2\sqrt{x} \right) dx$. Rearranging the terms: $I = \int e^{x} \left( 2\sqrt{x} + \frac{1}{\sqrt{x}} \right) dx$. Let $f(x) = 2\sqrt{x}$. Then $f'(x) = 2 \cdot \frac{1}{2\sqrt{x}} = \frac{1}{\sqrt{x}}$. Using the standard integral formula $\int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + c$,we get: $I = e^{x} (2\sqrt{x}) + c = 2\sqrt{x} e^{x} + c$.

$\int \frac{e^{x}}{\sqrt{x}}(1+2 x) d x=$

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