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Evaluate: \(\smallint \frac{{2x + 1}}{{\left( {x + 2} \right)\;\left( {x - 3} \right)}}dx\)
1. \(\frac{3}{5}\log \left| {x + 2} \right| - \frac{7}{5}\log \left| {x - 3} \right| + C\)
2. \(\frac{3}{5}\log \left| {x + 2} \right| + \frac{7}{5}\log \left| {x - 3} \right| + C\)
3. \(\frac{3}{5}\log \left| {x + 2} \right| + \frac{7}{5}\log \left| {x + 3} \right| + C\)
4. \(\frac{3}{5}\log \left| {x + 2} \right| - \frac{7}{5}\log \left| {x + 3} \right| + C\)
5. None of these

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Correct Answer - Option 2 : \(\frac{3}{5}\log \left| {x + 2} \right| + \frac{7}{5}\log \left| {x - 3} \right| + C\)

Concept:

Partial Fraction:

Factors in the denominator

Corresponding Partial Fraction

(x - a)

\(\frac{A}{{x - a}}\)

(x - b)2

\(\frac{A}{{x - b}} + \frac{B}{{{{\left( {x - b} \right)}^2}}}\)

(x - a) (x - b)

\(\frac{A}{{\left( {x - a} \right)}} + \frac{B}{{\left( {x - b} \right)}}\)

(x - c)3

\(\frac{A}{{x - c}} + \frac{B}{{{{\left( {x - c} \right)}^2}}} + \frac{C}{{{{\left( {x - c} \right)}^3}}}\)

(x - a) (x2 - a)

\(\frac{A}{{\left( {x - a} \right)}} + \frac{{Bx + C}}{{\left( {{x^2} - a} \right)}}\)

(ax2 + bx + c)

\(\frac{{Ax + B}}{{\left( {a{x^2} + bx + c} \right)}}\)


Calculation:

Here we have to find the value of \(\smallint \frac{{2x + 1}}{{\left( {x + 2} \right)\;\left( {x - 3} \right)}}dx\)

Let \(\frac{{2x + 1}}{{\left( {x + 2} \right)\;\left( {x - 3} \right)}} = \frac{A}{{\left( {x + 2} \right)}} + \frac{B}{{\left( {x - 3} \right)}}\)

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

By putting t = - 2 on both the sides of (1) we get A = 3/5

By putting t = 3 on both the sides of (1) we get B = 7/5

\(\Rightarrow \frac{{2x + 1}}{{\left( {x + 2} \right)\;\left( {x - 3} \right)}} = \frac{3}{{5\left( {x + 2} \right)}} + \frac{7}{{5\left( {x - 3} \right)}}\)

\(\Rightarrow \smallint \frac{{2x + 1}}{{\left( {x + 2} \right)\;\left( {x - 3} \right)}}dx = \frac{3}{5}\;\smallint \frac{{dx}}{{x + 2}} + \frac{7}{5}\;\smallint \frac{{dx}}{{x - 3}}\)

As we know  that \(\smallint \frac{{dx}}{x} = \log \left| x \right|\; + C\)  where C is a constant

\(\Rightarrow \smallint \frac{{2x + 1}}{{\left( {x + 2} \right)\;\left( {x - 3} \right)}}dx = \frac{3}{5}\log \left| {x + 2} \right| + \frac{7}{5}\log \left| {x - 3} \right| + C\)

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