Find \(\int \frac{8 dx}{x^2-16}\).

Category: QuestionsFind \(\int \frac{8 dx}{x^2-16}\).
Editor">Editor Staff asked 11 months ago

Find \(\int \frac{8 dx}{x^2-16}\).
 
(a) \(log⁡\left |\frac{4+x}{4-x}\right |+C\)
 
(b) –\(log⁡\left |\frac{4+x}{4-x}\right |+C\)
 
(c) \(8 log⁡\left |\frac{4+x}{4-x}\right |+C\)
 
(d) \(\frac{1}{8} log⁡\left |\frac{4+x}{4-x}\right |+C\)
 
The question was asked in class test.
 
Origin of the question is Integrals of Some Particular Functions in chapter Integrals of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

Correct option is (a) log⁡\left |\frac{4+x}{4-x}\right |+C
 
To elaborate: \int \frac{8dx}{16-x^2}=8\int \frac{dx}{4^2-x^2}
 
By using the formula \int \frac{dx}{a^2-x^2}=\frac{1}{2a} \left |\frac{a+x}{a-x}\right |+C
 
∴8\int \frac{dx}{4^2-x^2}=8(\frac{1}{2(4)} log⁡\left |\frac{4+x}{4-x}\right |)+8C_1
 
8\int \frac{dx}{4^2-x^2}=log⁡\left |\frac{4+x}{4-x}\right |+C