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MHTCET 17 Apr 2026 Shift 2 Paper

Question : 129 of 150

Marks: +1, -0

\int \frac{3 x+7}{x^{2}-3 x+2} d x=m \log \left(\frac{x-2}{x-1}\right)+n \log (x-2)+c

m, n \in R

c

m+n=

Solution:

Concept:

Split the integrand using partial fractions and compare the result with the given form.

Explanation:

Factor the denominator:

x^{2}-3x+2=(x-1)(x-2)

Write the partial fraction decomposition:

\frac{3x+7}{(x-1)(x-2)}=\frac{A}{x-1}+\frac{B}{x-2}

Multiplying by

(x-1)(x-2)

3x+7=A(x-2)+B(x-1)

Comparing coefficients:

A+B=3,\quad -2A-B=7

Solving gives:

A=-10,\quad B=13

Therefore:

\int \frac{3x+7}{x^{2}-3x+2}\,dx=-10\log|x-1|+13\log|x-2|+c

Rewrite the given form:

m\log\left(\frac{x-2}{x-1}\right)+n\log(x-2)=-m\log(x-1)+(m+n)\log(x-2)

Compare with

-10\log|x-1|+13\log|x-2|

-m=-10 \Rightarrow m=10

m+n=13

Answer:

m+n=13

Hence, the correct option is D.

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