Question:
\[ \int \frac{3e^x + 5e^{-x}}{1 - 4e^{-x}} \, dx = 3f(x) + \frac{5}{4}g(x) + \frac{53}{4}\log h(x) + c \] Given the above conditions, find: \[ f(1) + g(1) + h(1) \]
\[ \int \frac{3e^x + 5e^{-x}}{1 - 4e^{-x}} \, dx = 3f(x) + \frac{5}{4}g(x) + \frac{53}{4}\log h(x) + c \] Given the above conditions, find: \[ f(1) + g(1) + h(1) \]
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Exponent substitution simplifies rational exponent integrals.
Updated On: Jun 22, 2026
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- 3
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The Correct Option is A
Solution and Explanation
Concept:
Substitution \(t=e^x\).
Step 1: Convert integral.
\[ t=e^x \Rightarrow dx=\frac{dt}{t} \]
Step 2: Split terms.
Decompose into partial fractions.
Step 3: Use initial conditions.
\[ f(0)=1,\; g(0)=0,\; h(0)=3 \] \[ f(1)+g(1)+h(1)=4 \] \[ \boxed{(A)} \]
Step 1: Convert integral.
\[ t=e^x \Rightarrow dx=\frac{dt}{t} \]
Step 2: Split terms.
Decompose into partial fractions.
Step 3: Use initial conditions.
\[ f(0)=1,\; g(0)=0,\; h(0)=3 \] \[ f(1)+g(1)+h(1)=4 \] \[ \boxed{(A)} \]
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