Advertisements
Advertisements
प्रश्न
\[\int e^x \left( \log x + \frac{1}{x} \right) dx\]
योग
Advertisements
उत्तर
\[\text{ Let I }= \int e^x \left( \log x + \frac{1}{x} \right)dx\]
\[\text{ Here}, f(x) = \log x\]
\[ \Rightarrow f'(x) = \frac{1}{x}\]
\[\text{ put }\ e^x f(x) = t\]
\[ \Rightarrow e^x \log x = t\]
\[\text{ Diff both sides w . r . t x}\]
\[ e^x \log x + e^x \frac{1}{x} = \frac{dt}{dx}\]
\[ \Rightarrow e^x \left( \log x + \frac{1}{x} \right)dx = dt\]
\[ \therefore \int e^x \left[ \log x + \frac{1}{x} \right]dx = \int dt\]
\[ \Rightarrow I = t + C\]
\[ = e^x \log x + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{1}{\sqrt{x + a} + \sqrt{x + b}} dx\]
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
`∫ cos ^4 2x dx `
` ∫ cos mx cos nx dx `
\[\int\frac{1 - \sin x}{x + \cos x} dx\]
\[\int\frac{1 - \sin 2x}{x + \cos^2 x} dx\]
\[\int\sqrt {e^x- 1} \text{dx}\]
\[\int\left( 2 x^2 + 3 \right) \sqrt{x + 2} \text{ dx }\]
` ∫ 1 /{x^{1/3} ( x^{1/3} -1)} ` dx
\[\int \sec^4 2x \text{ dx }\]
\[\int \sin^5 x \text{ dx }\]
\[\int \sin^3 x \cos^5 x \text{ dx }\]
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 - 1}} dx\]
\[\int\frac{1}{2 x^2 - x - 1} dx\]
\[\int\frac{1}{\sqrt{7 - 3x - 2 x^2}} dx\]
\[\int\frac{x}{\sqrt{8 + x - x^2}} dx\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{1}{4 + 3 \tan x} dx\]
\[\int\frac{4 \sin x + 5 \cos x}{5 \sin x + 4 \cos x} \text{ dx }\]
\[\int\frac{\log \left( \log x \right)}{x} dx\]
\[\int\frac{\log x}{x^n}\text{ dx }\]
\[\int e^x \frac{1 + x}{\left( 2 + x \right)^2} \text{ dx }\]
\[\int\left( x - 2 \right) \sqrt{2 x^2 - 6x + 5} \text{ dx }\]
\[\int\frac{1}{\left( x + 1 \right)^2 \left( x^2 + 1 \right)} dx\]
\[\int\frac{x + 1}{x \left( 1 + x e^x \right)} dx\]
\[\int\frac{1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{x}{\left( x^2 + 2x + 2 \right) \sqrt{x + 1}} \text{ dx}\]
\[\int\frac{1}{1 + \tan x} dx =\]
\[\int\frac{\sin x}{3 + 4 \cos^2 x} dx\]
\[\int e^x \left\{ f\left( x \right) + f'\left( x \right) \right\} dx =\]
\[\int\frac{x + 2}{\left( x + 1 \right)^3} \text{ dx }\]
\[\int \sin^4 2x\ dx\]
\[\int\frac{1}{\left( \sin^{- 1} x \right) \sqrt{1 - x^2}} \text{ dx} \]
\[\int\frac{1}{e^x + e^{- x}} dx\]
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} \text{ dx }\]
\[\int\frac{1}{\sqrt{x^2 + a^2}} \text{ dx }\]
\[\int\frac{1}{\sec x + cosec x}\text{ dx }\]
\[\int x\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int\frac{\cot x + \cot^3 x}{1 + \cot^3 x} \text{ dx}\]
