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: Use repeated differentiation of exponential functions. Recall that \[ \frac{d}{dx}(a^{u}) = a^{u}\ln(a)\,\frac{du}{dx}. \]

Step 1: Differentiate \(4^{\,4^{4^x}}\). Let \[ y=4^{\,4^{4^x}}. \] Then \[ \frac{dy}{dx} = 4^{\,4^{4^x}}\ln4 \cdot \frac{d}{dx}\!\left(4^{4^x}\right). \] Now, \[ \frac{d}{dx}\!\left(4^{4^x}\right) = 4^{4^x}\ln4 \cdot \frac{d}{dx}(4^x). \] Also, \[ \frac{d}{dx}(4^x) = 4^x\ln4. \] Therefore, \[ \frac{dy}{dx} = 4^{\,4^{4^x}} (\ln4) \cdot 4^{4^x} (\ln4) \cdot 4^x (\ln4). \] \[ = (\ln4)^3 \, 4^x \, 4^{4^x} \, 4^{\,4^{4^x}}. \]

Step 2: Relate with the given integrand. Hence \[ 4^x \, 4^{4^x} \, 4^{\,4^{4^x}} = \frac{1}{(\ln4)^3} \frac{d}{dx} \left( 4^{\,4^{4^x}} \right). \] Therefore, \[ \int 4^x \, 4^{4^x} \, 4^{\,4^{4^x}} \,dx = \frac1{(\ln4)^3} \,4^{\,4^{4^x}} +C. \] Comparing with \[ A\,4^{\,4^{4^x}}+C, \] we get \[ A=\frac1{(\ln4)^3}. \]

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