Match List-I with List-II
[c is an arbitrary constant]
| List-I | List-II | ||
| (A). | \( I=\int 6x^{4}dx= \) | (I). | \( \log|x^{2}+3x+5|+c \) |
| (B). | \( I=\int\displaystyle\frac{2x+3}{x^{2}+3x+5}dx= \) | (II). | \( x^{3}e^{x}+c \) |
| (C). | \( I=\int\sqrt[3]{x}\,dx= \) | (III). | \( \displaystyle\frac{6}{5}x^{5}+c \) |
| (D). | \( I=\int e^{x}(x^{3}+3x^{2})\,dx= \) | (IV). | \( \displaystyle\frac{3}{4}x^{\frac{4}{3}}+c \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (III), (B) - (I), (C) - (IV), (D) - (II)
(A) \( \int 6x^{4}\,dx \)
This matches (III).
By the power rule,
\( \displaystyle \int 6x^{4}\,dx=\frac{6}{5}x^{5}+c \)
(B) \( \int\frac{2x+3}{x^{2}+3x+5}\,dx \)
This matches (I).
The numerator \( 2x+3 \) is the derivative of the denominator, so the integral is \( \log|x^{2}+3x+5|+c \).
(C) \( \int\sqrt[3]{x}\,dx \)
This matches (IV).
Writing \( \sqrt[3]{x}=x^{1/3} \),
\( \displaystyle \int x^{1/3}\,dx=\frac{x^{4/3}}{4/3}+c \)
\( \displaystyle =\frac{3}{4}x^{4/3}+c \)
(D) \( \int e^{x}(x^{3}+3x^{2})\,dx \)
This matches (II).
The integral has the form \( \int e^{x}[f(x)+f'(x)]\,dx=e^{x}f(x)+c \) with \( f(x)=x^{3} \), whose derivative is \( 3x^{2} \). So the value is \( x^{3}e^{x}+c \).
So the matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).