Question

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 \(f(1)\).

• A. \(\sqrt{3}\,e\)
• B. \(\sqrt{5}\,e\)
• C. \(\sqrt{2}\,e\)
• D. \(e\)
• E.

Hint:

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.

Answer 1

The Correct Option is C

Concept: To evaluate an integral containing \(e^x\) multiplied by another expression, we often check if the integrand is the derivative of a product involving \(e^x\). This helps simplify the integration directly.

Step 1: Observe the integrand structure. \[ f(x) = \int e^x \left(\frac{x^2 + x + 1}{\sqrt{x^2 + 1}}\right) dx \] Notice that the expression resembles the derivative of \[ e^x \sqrt{x^2+1} \]

Step 2: Differentiate \(e^x\sqrt{x^2+1}\). Using the product rule, \[ \frac{d}{dx}\left(e^x\sqrt{x^2+1}\right) = e^x\sqrt{x^2+1} + e^x\frac{x}{\sqrt{x^2+1}} \] \[ = e^x \left(\frac{x^2 + 1 + x}{\sqrt{x^2+1}}\right) \] \[ = e^x \left(\frac{x^2 + x + 1}{\sqrt{x^2+1}}\right) \] Thus, \[ f(x) = e^x\sqrt{x^2+1} + C \]

Step 3: Use the given condition \(f(0)=1\). \[ f(0) = e^0\sqrt{0^2+1} + C \] \[ 1 = 1 + C \] \[ C = 0 \] Thus, \[ f(x) = e^x\sqrt{x^2+1} \]

Step 4: Find \(f(1)\). \[ f(1) = e^1\sqrt{1^2+1} \] \[ f(1) = e\sqrt{2} \] Hence, \[ f(1) = \sqrt{2}\,e \]