The value of the integral ₁² x d x/(x+2)(x+3) is

Correct Answer is : log⁡(11251024) ≤ft(1125/1024)log(10241125)

Correct Answer is : log⁡(11251024) ≤ft(1125/1024)log(10241125)

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MHTCET 22 Apr 2025 Shift 2 Paper

Question : 146 of 150

Marks: +1, -0

\int_{1}^{2} \frac{x d x}{(x+2)(x+3)}

Solution:

Concept:

Use partial fractions to split the integrand, then integrate term by term and apply logarithm rules.

Explanation:

Let

I = \int_{1}^{2} \frac{x}{(x+2)(x+3)} \, dx

Write

\frac{x}{(x+2)(x+3)} = \frac{A}{x+2} + \frac{B}{x+3}

Then

x = A(x+3) + B(x+2)

Solving gives

A = -2

and

B = 3

I = \int_{1}^{2} \frac{-2}{x+2} \, dx + \int_{1}^{2} \frac{3}{x+3} \, dx

This equals

[-2 \log|x+2| + 3 \log|x+3|]_{1}^{2}

Substitute the limits:

-2(\log 4 - \log 3) + 3(\log 5 - \log 4)

Simplify:

\log\left(\frac{9}{16}\right) + \log\left(\frac{125}{64}\right) = \log\left(\frac{9 \times 125}{16 \times 64}\right) = \log\left(\frac{1125}{1024}\right)

Answer:

\log\left(\frac{1125}{1024}\right)

Hence, Option D is correct.

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