Question

\( \int x^7 (x^8 + 1)^{-3/4} \, dx = \)

• A. \( \frac{1}{2}\left(1+\frac{1}{x^8}\right)^{1/4} + C \)
• B. \( 4\left(1+\frac{1}{x^8}\right)^{1/4} + C \)
• C. \( (x^8+1)^{1/4} + C \)
• D. \( 4(x^8+1)^{1/4} + C \)
• E. \( \frac{1}{2}(x^8+1)^{1/4} + C \)

Hint:

Match derivative of inner function to simplify substitution quickly.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This integral is best solved using the method of u-substitution. The key is to notice that the derivative of the inner function \(x^8+1\) is \(8x^7\), and a factor of \(x^7\) is present in the integrand. This makes it a prime candidate for substitution.
Step 2: Key Formula or Approach:
1. Let \(u = x^8 + 1\). 2. Find the differential \(du = 8x^7 dx\). 3. Rearrange this to express \(x^7 dx\) in terms of `du`. 4. Substitute `u` and `du` into the integral, which will now be in terms of `u`. 5. Evaluate the simplified integral using the power rule for integration. 6. Substitute back \(u = x^8 + 1\) to get the final answer.
Step 3: Detailed Explanation:
The integral is: \[ \int x^7(x^8+1)^{-3/4} dx \] Let's choose our substitution: \[ u = x^8 + 1 \] Differentiating with respect to x, we get: \[ \frac{du}{dx} = 8x^7 \] \[ du = 8x^7 dx \] We have \(x^7 dx\) in our integral, so we can write: \[ x^7 dx = \frac{du}{8} \] Now substitute `u` and `du` into the integral: \[ \int (u)^{-3/4} \left(\frac{du}{8}\right) \] Take the constant \(\frac{1}{8}\) outside the integral: \[ \frac{1}{8} \int u^{-3/4} du \] Now apply the power rule for integration, \(\int u^n du = \frac{u^{n+1}}{n+1}\): \[ \frac{1}{8} \left[ \frac{u^{-3/4 + 1}}{-3/4 + 1} \right] + C \] \[ \frac{1}{8} \left[ \frac{u^{1/4}}{1/4} \right] + C = \frac{1}{8} [4u^{1/4}] + C = \frac{4}{8} u^{1/4} + C = \frac{1}{2} u^{1/4} + C \] Finally, substitute back \(u = x^8 + 1\): \[ \frac{1}{2}(x^8+1)^{1/4} + C \] Step 4: Final Answer:
The integral is \(\frac{1}{2}(x^8+1)^{1/4} + C\).