Question

If \[ \int 4^x \cdot 4^{4^x}\cdot 4^{\,4^{4^x}}\,dx = A\,4^{\,4^{4^x}}+C, \] then \(A=\)

• A. \[ \frac1{\ln 4} \]
• B. \[ \frac1{(\ln 4)^2} \]
• C. \[ \frac1{(\ln 4)^3} \]
• D. \[ \frac1{(\ln 4)^4} \]

Hint:

For nested exponentials such as \[ a^{a^{a^x}}, \] each differentiation contributes one factor of \[ \ln a. \] Count the number of exponential layers to determine the power of \(\ln a\).

Answer 1

The Correct Option is C

Concept: Recognize the integrand as the derivative of \(4^{4^{4^x}}\) up to a constant factor. Use the chain rule to find that constant, then integrate.

Step 1: Let \(F(x) = 4^{4^{4^x}}\). Then \(F'(x) = 4^{4^{4^x}} \ln 4 \cdot \frac{d}{dx}(4^{4^x}) = 4^{4^{4^x}} \ln 4 \cdot 4^{4^x} \ln 4 \cdot 4^x \ln 4 = (\ln 4)^3 4^x 4^{4^x} 4^{4^{4^x}}\).

Step 2: Rearranging, \(4^x 4^{4^x} 4^{4^{4^x}} = \frac{1}{(\ln 4)^3} F'(x)\). Integrate: \(\int 4^x 4^{4^x} 4^{4^{4^x}} dx = \frac{1}{(\ln 4)^3} 4^{4^{4^x}} + C\).

Step 3: Compare with \(A\,4^{4^{4^x}} + C\) to get \(A = \frac{1}{(\ln 4)^3}\).

Step 4: Write the final answer. \(\boxed{\frac1{(\ln4)^3}}\)