Question:
\( \int x^7 (x^8 + 1)^{-3/4} \, dx = \)
\( \int x^7 (x^8 + 1)^{-3/4} \, dx = \)
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Match derivative of inner function to simplify substitution quickly.
Updated On: Apr 21, 2026
- \( \frac{1}{2}\left(1+\frac{1}{x^8}\right)^{1/4} + C \)
- \( 4\left(1+\frac{1}{x^8}\right)^{1/4} + C \)
- \( (x^8+1)^{1/4} + C \)
- \( 4(x^8+1)^{1/4} + C \)
- \( \frac{1}{2}(x^8+1)^{1/4} + C \)
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The Correct Option is
Solution and Explanation
Concept:
Use substitution.
Step 1: Let \( u = x^8+1 \).
\[ du = 8x^7 dx \Rightarrow x^7 dx = \frac{du}{8} \]
Step 2: Substitute.
\[ \int x^7(x^8+1)^{-3/4}dx = \frac{1}{8}\int u^{-3/4}du \]
Step 3: Integrate.
\[ = \frac{1}{8}\cdot \frac{u^{1/4}}{1/4} = \frac{1}{2}u^{1/4} \] \[ = \frac{1}{2}(x^8+1)^{1/4} \]
Step 1: Let \( u = x^8+1 \).
\[ du = 8x^7 dx \Rightarrow x^7 dx = \frac{du}{8} \]
Step 2: Substitute.
\[ \int x^7(x^8+1)^{-3/4}dx = \frac{1}{8}\int u^{-3/4}du \]
Step 3: Integrate.
\[ = \frac{1}{8}\cdot \frac{u^{1/4}}{1/4} = \frac{1}{2}u^{1/4} \] \[ = \frac{1}{2}(x^8+1)^{1/4} \]
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