∫dx3x3xn+1 =3∫x2dxx3xn+1 =3∫dxxxn+1 =∫xn−1xnxn+1dxxn=t n xn-1dx=dt =3n∫dtt(t+1) =3n∫(t+1)−tt(t+1)dt =3n∫1t−1t+1dt=3n(ln⁡t−ln⁡(t+1))+c=3nln⁡tt+1+c

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$\int \frac{d x^{3}}{x^{3} (x^{n} + 1)} equals$

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$\frac{3}{n} \ln (\frac{x^{n}}{x^{n} + 1}) + c$

$\frac{1}{n} \ln (\frac{x^{n}}{x^{n} + 1}) + c$

$\frac{3}{n} \ln (\frac{x^{n} + 1}{x^{n}}) + c$

$3 n \ln (\frac{x^{n + 1}}{x^{n}}) + c$

answer is A.

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Detailed Solution

$\begin{array}{l} \int \frac{d x^{3}}{x^{3} (x^{n} + 1)} \\ = 3 \int \frac{x^{2} d x}{x^{3} (x^{n} + 1)} \\ = 3 \int \frac{d x}{x (x^{n} + 1)} \\ = \int \frac{x^{n - 1}}{x^{n} (x^{n} + 1)} d x \\ x^{n} = t \\ n x^{n - 1} d x = d t \\ = \frac{3}{n} \int \frac{d t}{t (t + 1)} \\ = \frac{3}{n} \int \frac{(t + 1) - t}{t (t + 1)} d t \\ = \frac{3}{n} \int (\frac{1}{t} - \frac{1}{t + 1}) d t \\ = \frac{3}{n} (\ln t - \ln (t + 1)) + c \\ = \frac{3}{n} \ln (\frac{t}{t + 1}) + c \end{array}$