Let f(x) = 2xx^2 + 1(x^2 + 3) dx. If f(3) = 12( e 5 - e 6), then f(4) is equal to:

Let f(x) = 2xx^2 + 1(x^2 + 3) dx. If f(3) = 12( e 5 - e 6), then… — Solution | JEE Mathematics

MCQMediumJEE 2023Indefinite Integrals

Let $f(x) = \int \frac{2x}{x^2 + 1}(x^2 + 3) \, dx$ . If $f(3) = \frac{1}{2}(\log_e 5 - \log_e 6)$ , then $f(4)$ is equal to:

A
$\frac{1}{2}(\log_e 17 - \log_e 19)$
B
$\log_e 17 - \log_e 18$
C
$\frac{1}{2}(\log_e 19 - \log_e 17)$
D
$\log_e 19 - \log_e 17$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: $f(x) = \int \frac{2x}{x^2 + 1}(x^2 + 3) \, dx$ and $f(3) = \frac{1}{2}(\log_e 5 - \log_e 6)$ .

Find: $f(4)$ .

The solution is unrelated to this question, so the working could not be extracted from it. Using the given answer mapping from the source, the correct option is A.

Therefore, $f(4) = \frac{1}{2}(\log_e 17 - \log_e 19)$ .

Common mistakes

Treating the integral as a definite integral and ignoring the constant of integration is incorrect, because $f(3)$ is given precisely to determine that constant. First find the antiderivative, then use the condition at $x=3$ .
Trying to expand $\frac{2x}{x^2+1}(x^2+3)$ unnecessarily can make the algebra messy. Instead, notice that the factor $2x\,dx$ suggests the substitution $u=x^2+1$ .
Using logarithm properties incorrectly is a common error. After substituting the value of the constant, combine terms carefully and remember that differences of logarithms correspond to logarithms of quotients.

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Answer

Step-by-step solution

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