Question:
The value of \(∫(5x-2)^3\) dx is:
The value of \(∫(5x-2)^3\) dx is:
Updated On: May 11, 2025
- 3(5x-2)+C; C is constant of integration.
- 15(5x-2)2+C; C is constant of integration.
- (5x-2)4+C; C is constant of integration.
- \(\frac{(5x-2)^4}{20}\)+C;C is constant of integration.
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The Correct Option is D
Solution and Explanation
To solve the integral \(∫(5x-2)^3\,dx\), we use the substitution method. Let \(u=5x-2\), then \(du=5\,dx\) or \(dx=\frac{du}{5}\). The integral becomes:
\[∫u^3\cdot\frac{1}{5}\,du\]
Simplifying, we have:
\[\frac{1}{5}∫u^3\,du\]
Integrate \(u^3\):
\[\frac{1}{5}\cdot\frac{u^4}{4}+C=\frac{u^4}{20}+C\]
Substitute back \(u=5x-2\):
\[\frac{(5x-2)^4}{20}+C\]
This matches the correct answer: \(\frac{(5x-2)^4}{20}+C; C\) is the constant of integration.
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