Question

The value of \(∫(5x-2)^3\) dx is:

• A. 3(5x-2)+C; C is constant of integration.
• B. 15(5x-2)²+C; C is constant of integration.
• C. (5x-2)⁴+C; C is constant of integration.
• D. \(\frac{(5x-2)^4}{20}\)+C;C is constant of integration.

Answer 1

The Correct Option is D

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.