Match List I with List II.
| List I | List II | ||
| A. | \(\int 5x^4\,dx\) | I. | \(\displaystyle\frac{e^{3x}}{3}+C\) |
| B. | \(\int e^{3x}\,dx\) | II. | \(3e^{3x}+C\) |
| C. | \(\int 9e^{3x}\,dx\) | III. | \(20x^3+C\) |
| D. | \(\int 60x^2\,dx\) | IV. | \(x^5+C\) |
Choose the correct answer from the options given below
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A-IV, B-I, C-II, D-III
Using the power rule \(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\) and \(\int e^{kx}\,dx = \frac{e^{kx}}{k} + C\):
(A) \(\int 5x^4\,dx\)
This matches (IV) \(x^5+C\).
\( \displaystyle \int 5x^4\,dx = \frac{5x^5}{5} + C = x^5 + C \)
(B) \(\int e^{3x}\,dx\)
This matches (I) \(\frac{e^{3x}}{3}+C\).
\( \displaystyle \int e^{3x}\,dx = \frac{e^{3x}}{3} + C \)
(C) \(\int 9e^{3x}\,dx\)
This matches (II) \(3e^{3x}+C\).
\( \displaystyle \int 9e^{3x}\,dx = \frac{9e^{3x}}{3} + C = 3e^{3x} + C \)
(D) \(\int 60x^2\,dx\)
This matches (III) \(20x^3+C\).
\( \displaystyle \int 60x^2\,dx = \frac{60x^3}{3} + C = 20x^3 + C \)
So the matching is A-IV, B-I, C-II, D-III.