Question

Evaluate \( \int 7^{7^{7x}} 7^{7x} dx = \)

• A. \( 7^{7^{7x}} (\log 7)^3 + C \)
• B. \( 7^{7^{7x}} (\log 7)^2 + C \)
• C. \( 7^{7^{7x}} (\log 7) + C \)
• D. \( 7^{7^{7x}} (\log 7)^3 + C \)

Hint:

When dealing with complex exponential integrals, start by applying substitution and then simplify the expression for easier integration.

Answer 1

The Correct Option is D

Step 1: Use substitution to simplify the integral.
Let \( u = 7^{7x} \). Then \( du = 7^{7x} \cdot \ln(7) \cdot 7x dx \). We rewrite the integral as: \[ \int 7^{7^{7x}} 7^{7x} dx = \int 7^{u} (\log 7)^3 dx \] This simplifies the integral, and we apply basic integration techniques.
Step 2: Apply the appropriate integration formula.
We find that the integral evaluates to \( 7^{7^{7x}} (\log 7)^3 + C \).
Step 3: Conclusion.
The correct answer is \( 7^{7^{7x}} (\log 7)^3 + C \).