Advertisements
Advertisements
प्रश्न
\[\int \log_{10} x\ dx\]
बेरीज
Advertisements
उत्तर
\[\int \log_{10} x\ dx\]
\[ = \int\frac{\log_e x}{\log_e 10} dx\]
\[ = \frac{1}{\log_e 10}\int 1_{II} \cdot \log_I x \text{ dx}\]
\[ = \frac{1}{\log_e 10}\left[ \log_e x\int1 \text{ dx} - \int\left\{ \frac{d}{dx}\left( \log_e x \right)\int1 \text{ dx} \right\}\text{ dx}\right]\]
\[ = \frac{1}{\log_e 10}\left[ \log_e x \cdot x - \int\frac{1}{x} \times x \text{ dx} \right]\]
\[ = \frac{1}{\log_e 10}\left[ x \log_e x - x \right] + C\]
\[ = \frac{1}{\log_e 10} \times x \left( \log_e x - 1 \right) + C\]
\[ = x \left( \log_e x - 1 \right) \cdot \log_{10} e + C\]
\[ = \int\frac{\log_e x}{\log_e 10} dx\]
\[ = \frac{1}{\log_e 10}\int 1_{II} \cdot \log_I x \text{ dx}\]
\[ = \frac{1}{\log_e 10}\left[ \log_e x\int1 \text{ dx} - \int\left\{ \frac{d}{dx}\left( \log_e x \right)\int1 \text{ dx} \right\}\text{ dx}\right]\]
\[ = \frac{1}{\log_e 10}\left[ \log_e x \cdot x - \int\frac{1}{x} \times x \text{ dx} \right]\]
\[ = \frac{1}{\log_e 10}\left[ x \log_e x - x \right] + C\]
\[ = \frac{1}{\log_e 10} \times x \left( \log_e x - 1 \right) + C\]
\[ = x \left( \log_e x - 1 \right) \cdot \log_{10} e + C\]
shaalaa.com
या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\]
\[\int\frac{x^5 + x^{- 2} + 2}{x^2} dx\]
\[\int\frac{\left( x^3 + 8 \right)\left( x - 1 \right)}{x^2 - 2x + 4} dx\]
\[\int\frac{1}{\left( 7x - 5 \right)^3} + \frac{1}{\sqrt{5x - 4}} dx\]
\[\int \left( e^x + 1 \right)^2 e^x dx\]
\[\int\sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}} dx\]
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
` ∫ {sec x "cosec " x}/{log ( tan x) }` dx
\[\int\frac{1}{x (3 + \log x)} dx\]
\[\int\frac{1 - \sin 2x}{x + \cos^2 x} dx\]
\[\int\frac{x \sin^{- 1} x^2}{\sqrt{1 - x^4}} dx\]
` ∫ x {tan^{- 1} x^2}/{1 + x^4} dx`
\[\int \tan^3 \text{2x sec 2x dx}\]
\[\int\frac{1}{\sqrt{x} + x} \text{ dx }\]
\[\int \cot^5 x \text{ dx }\]
` ∫ {1}/{a^2 x^2- b^2}dx`
\[\int\frac{\left( 1 - x^2 \right)}{x \left( 1 - 2x \right)} \text
{dx\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{1}{3 + 2 \cos^2 x} \text{ dx }\]
\[\int\frac{1}{\cos x \left( \sin x + 2 \cos x \right)} dx\]
`int"x"^"n"."log" "x" "dx"`
\[\int2 x^3 e^{x^2} dx\]
\[\int \tan^{- 1} \left( \sqrt{x} \right) \text{dx }\]
\[\int x \sin^3 x\ dx\]
\[\int\sqrt{3 - x^2} \text{ dx}\]
\[\int\frac{x^2 + x - 1}{x^2 + x - 6} dx\]
\[\int\frac{2 x^2 + 7x - 3}{x^2 \left( 2x + 1 \right)} dx\]
\[\int\frac{dx}{\left( x^2 + 1 \right) \left( x^2 + 4 \right)}\]
\[\int\frac{x^2 - 3x + 1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{1}{\left( x + 1 \right) \sqrt{x^2 + x + 1}} \text{ dx }\]
\[\int\frac{\sin^6 x}{\cos^8 x} dx =\]
\[\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x}\]
\[\int\sqrt{\sin x} \cos^3 x\ \text{ dx }\]
\[\int\sqrt{1 + 2x - 3 x^2}\text{ dx } \]
\[\int\frac{x^5}{\sqrt{1 + x^3}} \text{ dx }\]
\[\int \sec^{- 1} \sqrt{x}\ dx\]
\[\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx}\]
\[\int \cos^{- 1} \left( 1 - 2 x^2 \right) \text{ dx }\]
\[\int\frac{x \sin^{- 1} x}{\left( 1 - x^2 \right)^{3/2}} \text{ dx}\]
