KCET · Maths · Indefinite Integration
If \(\frac{1}{(3-5 x)(2+3 x)}=\frac{A}{3-5 x}+\frac{B}{2+3 x}\), then A : \(B\) is
- A \(2: 3\)
- B \(5: 3\)
- C \(3: 5\)
- D \(3: 2\)
Answer & Solution
Correct Answer
(B) \(5: 3\)
Step-by-step Solution
Detailed explanation
Given, \(\frac{1}{(3-5 x)(2+3 x)}=\frac{A}{(3-5 x)}+\frac{B}{(2+3 x)}\)
By partial fraction
\[
1=(3 A-5 B) x+(2 A+3 B)
\]
On comparing the like powers of ' \(x\) '
and \(\quad \begin{array}{rr}3 A-5 B=0 & \ldots \text { (i) } \\ 2 A+3 B=1 & \ldots \text { (ii) }\end{array}\)
On solving, we get \(\mathrm{A}=\frac{5}{19}, \mathrm{~B}=\frac{3}{19}\)
Hence, \(\quad \mathrm{A}: \mathrm{B}=5: 3\)
By partial fraction
\[
1=(3 A-5 B) x+(2 A+3 B)
\]
On comparing the like powers of ' \(x\) '
and \(\quad \begin{array}{rr}3 A-5 B=0 & \ldots \text { (i) } \\ 2 A+3 B=1 & \ldots \text { (ii) }\end{array}\)
On solving, we get \(\mathrm{A}=\frac{5}{19}, \mathrm{~B}=\frac{3}{19}\)
Hence, \(\quad \mathrm{A}: \mathrm{B}=5: 3\)
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