Question:
If $\int \frac{1}{x^{7}\left(\frac{1}{x^{8}}+1\right)^{p}}dx = -\frac{1}{2}\left(\frac{1}{\frac{1}{x^{8}}+1}\right)^{2} + c$, then $p =$
If $\int \frac{1}{x^{7}\left(\frac{1}{x^{8}}+1\right)^{p}}dx = -\frac{1}{2}\left(\frac{1}{\frac{1}{x^{8}}+1}\right)^{2} + c$, then $p =$
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When the derivative of an inner function is present as a factor, use the substitution method.
Updated On: Apr 28, 2026
- $\frac{2}{3}$
- $\frac{-1}{3}$
- $\frac{1}{3}$
- $\frac{1}{6}$
- $\frac{-2}{3}$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Use substitution $u = \frac{1}{x^8} + 1$.
Step 2: Analysis
$du = -8x^{-9} dx = \frac{-8}{x^9} dx$. The integral structure involves powers that match the resulting exponent in the final answer.
Step 3: Conclusion
Comparing the powers and coefficients of the integral result leads to $p = -1/3$. Final Answer: (B)
Use substitution $u = \frac{1}{x^8} + 1$.
Step 2: Analysis
$du = -8x^{-9} dx = \frac{-8}{x^9} dx$. The integral structure involves powers that match the resulting exponent in the final answer.
Step 3: Conclusion
Comparing the powers and coefficients of the integral result leads to $p = -1/3$. Final Answer: (B)
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