Let . If and , then is equal to: [2026]

Question

Let $f (x) = \int (\frac{16 x + 24}{x^{2} + 2 x - 15}) d x$ . If $f (4) = 14 \log_{e} (3)$ and $f (7) = \log_{e} (2^{α} \cdot 3^{β}), α, β \in N$ , then $α + β$ is equal to: [2026]

Smart Achievers · Accepted Answer

(3)

We have,

$f (x) = \int \frac{16 x + 24}{x^{2} + 2 x - 15} d x$

$= \int \frac{16 x + 24}{(x + 5) (x - 3)} d x$

$\frac{16 x + 24}{(x + 5) (x - 3)} = \frac{A}{x + 5} + \frac{B}{x - 3}$

$\Rightarrow$ 16x + 24 = A(x – 3) + B(x + 5)

$∴$ 16 = A + B and 24 = –3A + 5B

$\Rightarrow$ 24 = –3A + 5(16 – A) = –8A + 80

$∴ f (x) = \int (\frac{7}{x + 5} + \frac{9}{x - 3}) d x$

$\Rightarrow f (x) = 7 \log_{e} | x + 5 | + 9 \log_{e} | x - 3 | + c$

$f (4) = 7 \log_{e} | 9 | + 9 \log_{e} | 1 | + c$

$14 \log_{e} 3 = 7 \log_{e} | 9 | + c \Rightarrow c = 0$

$∴ f (7) = 7 \log_{e} 12 + 9 \log_{e} 4$

$= 32 \log_{e} 2 + 7 \log_{e} 3$

$= \log_{e} (2^{32} \cdot 3^{7})$

$∴ α + β = 39$

Solution

Q.
Let $f (x) = \int (\frac{16 x + 24}{x^{2} + 2 x - 15}) d x$ . If $f (4) = 14 \log_{e} (3)$ and $f (7) = \log_{e} (2^{α} \cdot 3^{β}), α, β \in N$ , then $α + β$ is equal to: [2026]

1 31

2 37

3 39

4 41

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