The correct option is B #10; 1nlog|x^n/x^n+1|+c #10; ∫1x(x^n+1)dx=∫1x^n+1(1+1x^n) 1x^n=t n x^ n 1dx=dt dxx^n+1=dt n 1n∫dt( ...

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Sum of Binomial Coefficients with Alternate Signs

∫1/xxn+1dx=

Question

$\int \frac{1}{x (x^{n} + 1)} d x =$

\frac{1}{n} log | \frac{x^{n}}{x^{n} + 1} | + c

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\frac{1}{n} log | \frac{x^{n + 1}}{x^{n}} | + c

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log | \frac{x^{n + 1}}{x^{n}} | + c

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log | \frac{x^{n}}{x^{n + 1}} | + c

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Solution

The correct option is B $\frac{1}{n} log | \frac{x^{n}}{x^{n} + 1} | + c$
$\int \frac{1}{x (x^{n} + 1)} d x = \int \frac{1}{x^{n} + 1 (1 + \frac{1}{x^{n}})}$
$\frac{1}{x^{n}} = t$
$- n x^{- n - 1} d x = d t$
$\frac{d x}{x^{n} + 1} = \frac{d t}{- n}$
$- \frac{1}{n} \int \frac{d t}{(t + 1)} = - \frac{1}{n} log ∣ t + 1 ∣ + c$
$= \frac{- 1}{n} log ∣ \frac{1}{x^{n}} + 1 ∣ + c$
$= \frac{- 1}{n} log ∣ \frac{x^{n} + 1}{x^{n}} ∣ + c$
$= \frac{1}{n} log ∣ \frac{x^{n}}{x^{n} + 1} ∣ + c$

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[Hint: multiply numerator and denominator by xⁿ^{− 1} and put xⁿ = t]

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