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Evaluate: \(\rm \int \frac{x+4}{(x+5)^2}dx\)1. \(\rm log\left | x+5 \right |-\frac{1}{(x+5)}+C\)2. \(\rm 2log\left | x+5 \right |+C\)3. \(\rm log\left | x+5 \right |+\frac{1}{(x+5)}+C\)4. \(\rm log\left | x+5 \right |+\frac{1}{(x+5)^2} + c\) |
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Answer» Correct Answer - Option 3 : \(\rm log\left | x+5 \right |+\frac{1}{(x+5)}+C\) Concept:
Calculation: \(\rm \int \frac{x+4}{(x+5)^2}dx\) This integrand is a proper rational fraction. So, by using the form of a partial fraction, we can write it as: ⇒\(\rm \frac{x+4}{(x+5)^2}=\frac{A}{x+5}+\frac{B}{(x+5)^2}\) This gives, x + 4 = Ax + 5A + B By comparing the coefficient of x and constant terms on both sides, we get A = 1 and 5A + B = 4 By solving these equation, we get A = 1 and B = -1 \(⇒ \rm \frac{x+4}{(x+5)^2}=\frac{1}{x+5}-\frac{1}{(x+5)^2}\) ⇒\(\rm \int \frac{x+4}{(x+5)^2}dx=\int\frac{1}{x+5}dx-\int\frac{1}{(x+5)^2}dx\) ⇒ \(\rm \int\frac{x+4}{(x+5)^2}dx=log\left | x+5 \right |+\frac{1}{(x+5)}+C\) Hence, option 3 is correct. |
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