Question:
If
\[
\int(e^{2x}+2e^x)\sqrt{e^{2x}-4e^x+5}\,dx
=
\frac13[f(x)]^{3/2}
+
4\left[
\frac{e^x-2}{2}\sqrt{f(x)}
+
\frac12g(x)
\right]+c
\]
then \(f(0)=\)
If
\[
\int(e^{2x}+2e^x)\sqrt{e^{2x}-4e^x+5}\,dx
=
\frac13[f(x)]^{3/2}
+
4\left[
\frac{e^x-2}{2}\sqrt{f(x)}
+
\frac12g(x)
\right]+c
\]
then \(f(0)=\)
Show Hint
In integration pattern questions, first identify repeated expression hidden inside the final answer structure.
Updated On: Jun 15, 2026
- \(2\)
- \(0\)
- \(1\)
- \(3\)
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
Identify repeated expression under square root.
Step 1: Observe integrand.
Inside root: \[ e^{2x}-4e^x+5 \] Thus naturally \[ f(x)=e^{2x}-4e^x+5 \]
Step 2: Evaluate at \(x=0\).
\[ f(0) = e^0-4e^0+5 \] \[ = 1-4+5 \] \[ =2 \] Thus \[ \boxed{2} \]
Step 1: Observe integrand.
Inside root: \[ e^{2x}-4e^x+5 \] Thus naturally \[ f(x)=e^{2x}-4e^x+5 \]
Step 2: Evaluate at \(x=0\).
\[ f(0) = e^0-4e^0+5 \] \[ = 1-4+5 \] \[ =2 \] Thus \[ \boxed{2} \]
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