Advertisements
Advertisements
प्रश्न
\[\int\frac{\log \left( \log x \right)}{x} \text{ dx}\]
बेरीज
Advertisements
उत्तर
\[\text{ Let I} = \int\frac{\log \left( \log x \right) dx}{x}\]
\[\text{ Putting log x = t}\]
\[ \Rightarrow \frac{1}{x} dx = dt\]
\[ \therefore I = \int 1_{II} \cdot \log _I t \cdot \text{ dt}\]
\[ = \log t\int1\text{ dt }- \int\left\{ \frac{d}{dt}\left( \log t \right)\int1 dt \right\}dt\]
\[ = \log t \cdot t - \int\frac{1}{t} \times t\text{ dt}\]
\[ = \log t \cdot t - \int dt\]
\[ = \log t \cdot t - t + C\]
\[ = t \left( \log t - 1 \right) + C\]
\[ = \log x \left( \text{ log} \left( \log x \right) - 1 \right) + C\]
shaalaa.com
या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
\[\int\frac{\sin^2 x}{1 + \cos x} \text{dx} \]
\[\int\frac{1}{1 - \cos x} dx\]
\[\int\frac{1}{1 - \sin x} dx\]
\[\int\frac{1}{\text{cos}^2\text{ x }\left( 1 - \text{tan x} \right)^2} dx\]
\[\int\left( x + 2 \right) \sqrt{3x + 5} \text{dx} \]
\[\int \text{sin}^2 \left( 2x + 5 \right) \text{dx}\]
\[\int\frac{\text{sin} \left( x - a \right)}{\text{sin}\left( x - b \right)} dx\]
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
\[\int\frac{1 - \sin 2x}{x + \cos^2 x} dx\]
\[\int\frac{\sin 2x}{\sin 5x \sin 3x} dx\]
\[\int\frac{x^5}{\sqrt{1 + x^3}} dx\]
\[\int\frac{1}{x^{2/3} \sqrt{x^{2/3} - 4}} dx\]
\[\int\frac{\cos x}{\sqrt{\sin^2 x - 2 \sin x - 3}} dx\]
\[\int\frac{x + 1}{x^2 + x + 3} dx\]
\[\int\frac{a x^3 + bx}{x^4 + c^2} dx\]
\[\int\frac{6x - 5}{\sqrt{3 x^2 - 5x + 1}} \text{ dx }\]
\[\int\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int\frac{1}{5 + 4 \cos x} dx\]
\[\int\frac{1}{p + q \tan x} \text{ dx }\]
\[\int\frac{2 \tan x + 3}{3 \tan x + 4} \text{ dx }\]
\[\int {cosec}^3 x\ dx\]
\[\int\left( x + 1 \right) \text{ e}^x \text{ log } \left( x e^x \right) dx\]
\[\int x^3 \tan^{- 1}\text{ x dx }\]
\[\int\left\{ \tan \left( \log x \right) + \sec^2 \left( \log x \right) \right\} dx\]
\[\int(2x + 5)\sqrt{10 - 4x - 3 x^2}dx\]
\[\int\frac{x^3}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx\]
\[\int\frac{5 x^2 - 1}{x \left( x - 1 \right) \left( x + 1 \right)} dx\]
\[\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} dx\]
Evaluate the following integral:
\[\int\frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)}dx\]
\[\int\frac{x^2 + 1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{1}{e^x + e^{- x}} dx\]
\[\int\frac{\sin x + \cos x}{\sqrt{\sin 2x}} \text{ dx}\]
\[\int \tan^4 x\ dx\]
\[\int\frac{x^2}{\left( x - 1 \right)^3} dx\]
\[\int\sqrt{\sin x} \cos^3 x\ \text{ dx }\]
\[\int\sqrt{a^2 - x^2}\text{ dx }\]
\[\int\frac{\log \left( 1 - x \right)}{x^2} \text{ dx}\]
\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]
