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 $p =$

• A. $\frac{2}{3}$
• B. $\frac{-1}{3}$
• C. $\frac{1}{3}$
• D. $\frac{1}{6}$
• E. $\frac{-2}{3}$

Hint:

When the derivative of an inner function is present as a factor, use the substitution method.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem can be solved by either performing the integration on the left side and comparing the result, or by differentiating the right side and comparing it to the integrand on the left. Differentiating is often easier.
Step 2: Key Formula or Approach:
Let \( F(x) = \frac{-1}{2(x^p+1)^2} + c \). We will find \( F'(x) \) and compare it to the integrand \( f(x) = \frac{x^{-4/3}}{(x^{-1/3}+1)^3} \). The Chain Rule for differentiation will be used: \( \frac{d}{dx} g(h(x)) = g'(h(x)) \cdot h'(x) \).
Step 3: Detailed Explanation:
Let's differentiate the right-hand side of the given equation with respect to x.
Let \( y = \frac{-1}{2}(x^p+1)^{-2} \).
\[ \frac{dy}{dx} = \frac{-1}{2} \cdot (-2)(x^p+1)^{-3} \cdot \frac{d}{dx}(x^p+1) \] \[ \frac{dy}{dx} = (x^p+1)^{-3} \cdot (px^{p-1}) \] \[ \frac{dy}{dx} = \frac{px^{p-1}}{(x^p+1)^3} \] This derivative must be equal to the integrand on the left side:
\[ \frac{px^{p-1}}{(x^p+1)^3} = \frac{x^{-4/3}}{(x^{-1/3}+1)^3} \] By comparing the denominators, we can equate the terms inside the parentheses:
\[ x^p = x^{-1/3} \] This implies that \( p = -\frac{1}{3} \).
Let's check if this value of p also makes the numerators match. Substitute \( p = -1/3 \) into the numerator of our derivative:
\[ px^{p-1} = (-\frac{1}{3})x^{-1/3 - 1} = -\frac{1}{3}x^{-4/3} \] So our derivative is \( \frac{-\frac{1}{3}x^{-4/3}}{(x^{-1/3}+1)^3} \). This doesn't match the integrand \( \frac{x^{-4/3}}{(x^{-1/3}+1)^3} \) due to the constant factor \( -1/3 \). This indicates a likely typo in the constants of the original problem statement. However, comparing the powers of x is the standard method for determining the unknown exponent 'p'.
Step 4: Final Answer:
By comparing the powers of x in the denominator of the integrand and the differentiated result, we find that \( p = -\frac{1}{3} \).