Advertisements
Advertisements
Question
\[\int\frac{\log\left( 1 + \frac{1}{x} \right)}{x \left( 1 + x \right)} dx\]
Sum
Advertisements
Solution
\[\int\frac{\log \left( 1 + \frac{1}{x} \right)}{x\left( 1 + x \right)}dx\]
\[Let, \log \left( 1 + \frac{1}{x} \right) = t\]
\[ \Rightarrow \frac{1}{1 + \frac{1}{x}} \times \frac{- 1}{x^2} = \frac{dt}{dx}\]
\[ \Rightarrow \left( \frac{x}{x + 1} \right) \times \frac{- 1}{x^2} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{- dx}{x\left( x + 1 \right)} = dt\]
\[ \Rightarrow \frac{dx}{x\left( x + 1 \right)} = - dt\]
\[Now, \int\frac{\log \left( 1 + \frac{1}{x} \right)}{x\left( 1 + x \right)}dx\]
= ∫ t . (-dt)
\[ = \frac{- t^2}{2} + C\]
\[ = - \frac{1}{2} \left\{ \log\left( 1 + \frac{1}{x} \right) \right\}^2 + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]
\[\int\frac{x^3 - 3 x^2 + 5x - 7 + x^2 a^x}{2 x^2} dx\]
\[\int\frac{1 + \cos x}{1 - \cos x} dx\]
\[\int\left( 5x + 3 \right) \sqrt{2x - 1} dx\]
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
` = ∫ root (3){ cos^2 x} sin x dx `
\[\int\frac{e^\sqrt{x} \cos \left( e^\sqrt{x} \right)}{\sqrt{x}} dx\]
\[\int\frac{\left( x + 1 \right) e^x}{\sin^2 \left( \text{x e}^x \right)} dx\]
\[\int\frac{\sin \left( \text{log x} \right)}{x} dx\]
\[\int\frac{x}{\sqrt{x^2 + a^2} + \sqrt{x^2 - a^2}} dx\]
\[\int\frac{1}{\left( x + 1 \right)\left( x^2 + 2x + 2 \right)} dx\]
Evaluate the following integrals:
\[\int\cos\left\{ 2 \cot^{- 1} \sqrt{\frac{1 + x}{1 - x}} \right\}dx\]
\[\int\frac{1}{\sqrt{8 + 3x - x^2}} dx\]
\[\int\frac{\sin 8x}{\sqrt{9 + \sin^4 4x}} dx\]
\[\int\frac{\left( 3 \sin x - 2 \right) \cos x}{5 - \cos^2 x - 4 \sin x} dx\]
\[\int\frac{x + 1}{\sqrt{x^2 + 1}} dx\]
\[\int\frac{1}{1 - \sin x + \cos x} \text{ dx }\]
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
\[\int\frac{2 \tan x + 3}{3 \tan x + 4} \text{ dx }\]
\[\int x^2 \text{ cos x dx }\]
\[\int e^\sqrt{x} \text{ dx }\]
\[\int\frac{x + \sin x}{1 + \cos x} \text{ dx }\]
\[\int\frac{x^2 \tan^{- 1} x}{1 + x^2} \text{ dx }\]
\[\int \cos^{- 1} \left( 4 x^3 - 3x \right) \text{ dx }\]
\[\int e^x \left( \tan x - \log \cos x \right) dx\]
\[\int e^x \left( \frac{x - 1}{2 x^2} \right) dx\]
\[\int x\sqrt{x^4 + 1} \text{ dx}\]
\[\int\sqrt{x^2 - 2x} \text{ dx}\]
\[\int\frac{x^2 + 1}{x^2 - 1} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{5 x^2 - 1}{x \left( x - 1 \right) \left( x + 1 \right)} dx\]
\[\int\frac{1}{1 + x + x^2 + x^3} dx\]
\[\int\frac{\left( x - 1 \right)^2}{x^4 + x^2 + 1} \text{ dx}\]
\[\int\frac{1}{\left( x^2 + 1 \right) \sqrt{x}} \text{ dx }\]
\[\int\frac{\sin^2 x}{\cos^4 x} dx =\]
\[\int\frac{1}{x^2 + 4x - 5} \text{ dx }\]
\[\int\frac{1}{\sqrt{3 - 2x - x^2}} \text{ dx}\]
\[\int\frac{5x + 7}{\sqrt{\left( x - 5 \right) \left( x - 4 \right)}} \text{ dx }\]
\[\int x \sec^2 2x\ dx\]
\[\int\frac{x^2}{\sqrt{1 - x}} \text{ dx }\]
