\(\int (5x - 2)^7\,dx\) is equal to (where C is a constant of integration)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{40}(5x - 2)^8 + C\)
Let \(t = 5x - 2\). Then
\(dt = 5\,dx\)
\(dx = \frac{1}{5}\,dt\)
So the integral becomes
\( \displaystyle \int t^7 \cdot \frac{1}{5}\,dt = \frac{1}{5} \cdot \frac{t^8}{8} + C \)
\( \displaystyle = \frac{1}{40}(5x - 2)^8 + C \)
The factor \(\frac{1}{5}\) comes from the coefficient of x inside the bracket.
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{8}(5x - 2)^8 + C\) — Incorrect. Its derivative is \(5(5x - 2)^7\), five times the integrand; the division by 5 is missing.
Option (2) \(\displaystyle\frac{5}{8}(5x - 2)^8 + C\) — Incorrect. Here the 5 is multiplied instead of divided; its derivative is \(25(5x - 2)^7\).
Option (4) \(35(5x - 2)^6 + C\) — Incorrect. This is close to the derivative of \((5x - 2)^7\), not its integral.