Question:
Evaluate \( \int 7^{7^{7x}} 7^{7x} dx = \)
Evaluate \( \int 7^{7^{7x}} 7^{7x} dx = \)
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When dealing with complex exponential integrals, start by applying substitution and then simplify the expression for easier integration.
Updated On: Jan 27, 2026
- \( 7^{7^{7x}} (\log 7)^3 + C \)
- \( 7^{7^{7x}} (\log 7)^2 + C \)
- \( 7^{7^{7x}} (\log 7) + C \)
- \( 7^{7^{7x}} (\log 7)^3 + C \)
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The Correct Option is D
Solution and Explanation
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 \( 7^{7^{7x}} (\log 7)^3 + C \).
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 \( 7^{7^{7x}} (\log 7)^3 + C \).
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