Question:medium

If \( f(x) = \displaystyle \int e^x \left(\frac{x^2 + x + 1}{\sqrt{x^2 + 1}}\right) dx \) such that the value of the function is \(1\) when \(x\) vanishes, find the value of \(f(1)\).

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When an integral contains \(e^x\) multiplied by another expression, check if the integrand matches the derivative of a product like \(e^x g(x)\). This often simplifies the integration immediately.

Updated On: May 2, 2026

\(\sqrt{3}\,e\)
\(\sqrt{5}\,e\)
\(\sqrt{2}\,e\)
\(e\)

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The Correct Option is C

Solution and Explanation

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If \( f(x) = \displaystyle \int e^x \left(\frac{x^2 + x + 1}{\sqrt{x^2 + 1}}\right) dx \) such that the value of the function is \(1\) when \(x\) vanishes, find the value of \(f(1)\).

Show Hint

The Correct Option is C

Solution and Explanation

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