Question:
The value of the integral $\int_1^2 \frac{x \text{ d}x}{(x+2)(x+3)}$ is
The value of the integral $\int_1^2 \frac{x \text{ d}x}{(x+2)(x+3)}$ is
Show Hint
Partial fractions: $\frac{x}{(x+a)(x+b)} = \frac{1}{b-a} (\frac{b}{x+b} - \frac{a}{x+a})$.
Updated On: Apr 26, 2026
- $\log \left( \frac{125}{16} \right)$
- $\log \left( \frac{1024}{1125} \right)$
- $\log \left( \frac{16}{125} \right)$
- $\log \left( \frac{1125}{1024} \right)$
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Partial Fractions
$\frac{x}{(x+2)(x+3)} = \frac{-2}{x+2} + \frac{3}{x+3}$.
Step 2: Integration
$[-2 \log(x+2) + 3 \log(x+3)]_1^2 = [\log \frac{(x+3)^3}{(x+2)^2}]_1^2$.
Step 3: Calculation
Value $= \log(\frac{5^3}{4^2}) - \log(\frac{4^3}{3^2}) = \log(\frac{125}{16}) - \log(\frac{64}{9}) = \log(\frac{125}{16} \times \frac{9}{64}) = \log(\frac{1125}{1024})$.
Wait, checking options... result is $\log(1125/1024)$.
Final Answer: (D)
$\frac{x}{(x+2)(x+3)} = \frac{-2}{x+2} + \frac{3}{x+3}$.
Step 2: Integration
$[-2 \log(x+2) + 3 \log(x+3)]_1^2 = [\log \frac{(x+3)^3}{(x+2)^2}]_1^2$.
Step 3: Calculation
Value $= \log(\frac{5^3}{4^2}) - \log(\frac{4^3}{3^2}) = \log(\frac{125}{16}) - \log(\frac{64}{9}) = \log(\frac{125}{16} \times \frac{9}{64}) = \log(\frac{1125}{1024})$.
Wait, checking options... result is $\log(1125/1024)$.
Final Answer: (D)
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Differentiation Questions
- If \( x^y = e^{x-y} \), at \( x = 1 \), find \( \frac{dy}{dx} \)?
- If \( y = \sec(\tan^{-1} x) \), find \( \frac{dy}{dx} \), given that \( x = 1 \):
- If \( y = e^{\sin(\cosec^{-1}x)} \), then \( \dfrac{dy}{dx} \) is
- If \( f(x) = e^{x}g(x) \), \( g(0) = 4 \), and \( g'(0) = 2 \), then \( f'(0) = \)
- If \( \frac{x}{x-y} = \log \left( \frac{a}{x-y} \right) \), then \( \frac{dy}{dx} = \)
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions