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Using method of partial fraction to evaluate ∫ (x + 5) dx/(x + 1) (x + 2)2 and the final answer becomes
- a)4 log (x + 1) – 4 log (x + 2) + 3/x + 2 + k
- b)4 log (x + 2) – 3/x + 2) + k
- c)4 log (x + 1) – 4 log (x + 2)
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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Using method of partial fraction to evaluate ∫ (x + 5) dx/(x + 1...
Given expression: $\frac{x+5}{(x+1)(x+2)^2}$
Step 1: Factorize the denominator
Denominator = $(x+1)(x+2)^2$
Step 2: Write the partial fraction
$\frac{x+5}{(x+1)(x+2)^2}=\frac{A}{x+1}+\frac{B}{x+2}+\frac{C}{(x+2)^2}$
Step 3: Find the values of A,B and C
Multiplying both sides by the denominator, we get:
$x+5=A(x+2)^2+B(x+1)(x+2)+C(x+1)$
Putting $x=-2$, we get:
$3A=3$
$\Rightarrow A=1$
Putting $x=-1$, we get:
$4B=6$
$\Rightarrow B=\frac{3}{2}$
Putting $x=-1$ again, we get:
$C=-\frac{1}{4}$
Step 4: Substitute the values of A,B and C in the partial fraction
$\frac{x+5}{(x+1)(x+2)^2}=\frac{1}{x+1}+\frac{3/2}{x+2}-\frac{1/4}{(x+2)^2}$
Step 5: Integrate the partial fraction
$\int \frac{x+5}{(x+1)(x+2)^2}dx=\int \frac{1}{x+1}dx+\frac{3}{2}\int \frac{1}{x+2}dx-\frac{1}{4}\int \frac{1}{(x+2)^2}dx$
$=ln|x+1|+\frac{3}{2}ln|x+2|+\frac{1}{4(x+2)}+k$
Step 6: Simplify the final answer
$=4ln|x+1|+4ln|x+2|-\frac{3}{x+2}+k$
Hence, the correct option is (a) $4ln|x+1|+4ln|x+2|-\frac{3}{x+2}+k$.
Step 1: Factorize the denominator
Denominator = $(x+1)(x+2)^2$
Step 2: Write the partial fraction
$\frac{x+5}{(x+1)(x+2)^2}=\frac{A}{x+1}+\frac{B}{x+2}+\frac{C}{(x+2)^2}$
Step 3: Find the values of A,B and C
Multiplying both sides by the denominator, we get:
$x+5=A(x+2)^2+B(x+1)(x+2)+C(x+1)$
Putting $x=-2$, we get:
$3A=3$
$\Rightarrow A=1$
Putting $x=-1$, we get:
$4B=6$
$\Rightarrow B=\frac{3}{2}$
Putting $x=-1$ again, we get:
$C=-\frac{1}{4}$
Step 4: Substitute the values of A,B and C in the partial fraction
$\frac{x+5}{(x+1)(x+2)^2}=\frac{1}{x+1}+\frac{3/2}{x+2}-\frac{1/4}{(x+2)^2}$
Step 5: Integrate the partial fraction
$\int \frac{x+5}{(x+1)(x+2)^2}dx=\int \frac{1}{x+1}dx+\frac{3}{2}\int \frac{1}{x+2}dx-\frac{1}{4}\int \frac{1}{(x+2)^2}dx$
$=ln|x+1|+\frac{3}{2}ln|x+2|+\frac{1}{4(x+2)}+k$
Step 6: Simplify the final answer
$=4ln|x+1|+4ln|x+2|-\frac{3}{x+2}+k$
Hence, the correct option is (a) $4ln|x+1|+4ln|x+2|-\frac{3}{x+2}+k$.
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Using method of partial fraction to evaluate ∫ (x + 5) dx/(x + 1) (x + 2)2 and the final answer becomesa)4 log (x + 1) – 4 log (x + 2) + 3/x + 2 + kb)4 log (x + 2) – 3/x + 2) + kc)4 log (x + 1) – 4 log (x + 2)d)none of theseCorrect answer is option 'A'. Can you explain this answer?
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