First slide

Evaluation of definite integrals

Question

$\int \frac{d x}{x (x^{n} + 1)} is equal to$

Moderate

A

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

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

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

none of these

Solution

$\begin{array}{l} I = \int \frac{d x}{x (x^{n} + 1)} = \int \frac{x^{n - 1}}{x^{n} (x^{n} + 1)} d x \\ Putting x^{n} = t so that n x^{n - 1} d x = d t, i . e . \\ We get x^{n - 1} d x = \frac{1}{n} d t \\ I = \int \frac{\frac{1}{n} d t}{t (t + 1)} = \frac{1}{n} \int (\frac{1}{t} - \frac{1}{t + 1}) d t \\ = \frac{1}{n} (\log t - \log (t + 1)) + C \\ == \frac{1}{n} \log (\frac{x^{n}}{x^{n} + 1}) + C \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App