{e}^{{x}}≤ft({2 {x}+1}{2 {{x}}}) {dx}=

Correct Answer is : exx+C e^{x} {x}+Cexx+C

Correct Answer is : exx+C e^{x} {x}+Cexx+C

Test Index

AP DSC School Assistant Math Model Test 3

Section: Subject Specific

Question : 56 of 160

Marks: +1, -0

\int \mathrm{e}^{\mathrm{x}}\left(\frac{2 \mathrm{x}+1}{2 \sqrt{\mathrm{x}}}\right) \mathrm{dx}=

Solution:

Concept:

The integral can be evaluated by recognizing that the integrand is the derivative of a product of functions. Specifically, the derivative of

e^{x}\sqrt{x}

gives the given expression.

Explanation:

Consider the function

f(x) = e^{x}\sqrt{x}

. Differentiate it using the product rule:

f'(x) = e^{x} \cdot \sqrt{x} + e^{x} \cdot \frac{1}{2\sqrt{x}} = e^{x}\left(\sqrt{x} + \frac{1}{2\sqrt{x}}\right)

Rewrite

\sqrt{x}

\frac{2x}{2\sqrt{x}}

f'(x) = e^{x}\left(\frac{2x}{2\sqrt{x}} + \frac{1}{2\sqrt{x}}\right) = e^{x}\left(\frac{2x+1}{2\sqrt{x}}\right)

Thus the integrand is exactly

f'(x)

. Therefore,

\int e^{x}\left(\frac{2x+1}{2\sqrt{x}}\right) dx = e^{x}\sqrt{x} + C

Answer:

Option D:

\rm e^{x} \sqrt{x}+C

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