Question:
If \( \int f(x)\,dx = g(x) \), then \( \int x^9 f(x^5)\,dx \) is equal to
If \( \int f(x)\,dx = g(x) \), then \( \int x^9 f(x^5)\,dx \) is equal to
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When \( f(x^n) \) appears $\longrightarrow$ try substitution \( t = x^n \) + integration by parts.
Updated On: Apr 22, 2026
- \( \frac{1}{5}(x^5 g(x^9) - 4\int g(x^9)\,dx) + C \)
- \( \frac{1}{5}[x^9 g(x^5) - \frac{1}{5}\int x^4 g(x^5)\,dx] + C \)
- \( \frac{1}{5}[g(x^9) + \int g(x^5)\,dx] \)
- \( \frac{x^5}{5} g(x^5) - \int x^4 g(x^5)\,dx + C \)
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The Correct Option is D
Solution and Explanation
Concept:
Use substitution:
\[
t = x^5
\Rightarrow dt = 5x^4 dx
\]
Step 1: Rewrite integral.
\[ \int x^9 f(x^5)\,dx = \int x^5 \cdot x^4 f(x^5)\,dx \]
Step 2: Substitute.
\[ = \int x^5 f(t) \cdot \frac{dt}{5} = \frac{1}{5} \int t f(t)\,dt \]
Step 3: Use integration by parts.
\[ \int t f(t)\,dt = t g(t) - \int g(t)\,dt \]
Step 4: Back substitute.
\[ = \frac{x^5}{5} g(x^5) - \int x^4 g(x^5)\,dx \]
Step 1: Rewrite integral.
\[ \int x^9 f(x^5)\,dx = \int x^5 \cdot x^4 f(x^5)\,dx \]
Step 2: Substitute.
\[ = \int x^5 f(t) \cdot \frac{dt}{5} = \frac{1}{5} \int t f(t)\,dt \]
Step 3: Use integration by parts.
\[ \int t f(t)\,dt = t g(t) - \int g(t)\,dt \]
Step 4: Back substitute.
\[ = \frac{x^5}{5} g(x^5) - \int x^4 g(x^5)\,dx \]
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