Advertisements
Advertisements
प्रश्न
\[\int x^2 e^{- x} \text{ dx }\]
बेरीज
Advertisements
उत्तर
\[\int x^2 e^{- x} \text{ dx }\]
` " Taking x"^2" as the first function and e"^- x " as the second function ".`
\[ = x^2 \int e^{- x} dx - \int\left( \frac{d}{dx} x^2 \int e^{- x} dx \right)dx\]
\[ = - x^2 e^{- x} - \int2x\left( e^{- x} \right)\left( - 1 \right)dx\]
\[ = - x^2 e^{- x} + 2\int x e^{- x} dx\]
\[ = - x^2 e^{- x} + 2\left[ - x e^{- x} + \int e^{- x} dx \right]\]
\[ = - x^2 e^{- x} + 2\left[ - x e^{- x} - e^{- x} \right] + C\]
\[ = - e^{- x} \left[ x^2 + 2x + 2 \right] + C\]
shaalaa.com
या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
APPEARS IN
संबंधित प्रश्न
\[\int\left\{ x^2 + e^{\log x}+ \left( \frac{e}{2} \right)^x \right\} dx\]
\[\int\frac{\left( x^3 + 8 \right)\left( x - 1 \right)}{x^2 - 2x + 4} dx\]
\[\int\frac{3x + 5}{\sqrt{7x + 9}} dx\]
\[\int \text{sin}^2 \left( 2x + 5 \right) \text{dx}\]
` ∫ cos mx cos nx dx `
\[\int\frac{\cos^3 x}{\sqrt{\sin x}} dx\]
\[\int x^3 \sin x^4 dx\]
\[\int\left( 2 x^2 + 3 \right) \sqrt{x + 2} \text{ dx }\]
\[\int \cot^5 x \text{ dx }\]
` ∫ {1}/{a^2 x^2- b^2}dx`
\[\int\frac{1}{4 x^2 + 12x + 5} dx\]
\[\int\frac{x + 2}{2 x^2 + 6x + 5}\text{ dx }\]
\[\int\frac{1}{1 + 3 \sin^2 x} \text{ dx }\]
\[\int\frac{1}{5 + 4 \cos x} dx\]
\[\int\frac{2 \tan x + 3}{3 \tan x + 4} \text{ dx }\]
\[\int x^3 \cos x^2 dx\]
\[\int \sin^{- 1} \sqrt{x} \text{ dx }\]
\[\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}} dx\]
\[\int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx\]
\[\int e^x \frac{x - 1}{\left( x + 1 \right)^3} \text{ dx }\]
\[\int\sqrt{3 - x^2} \text{ dx}\]
\[\int\frac{1}{1 + x + x^2 + x^3} dx\]
\[\int\frac{\left( x^2 + 1 \right) \left( x^2 + 2 \right)}{\left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx\]
\[\int\frac{1}{\left( 1 + x^2 \right) \sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{\cos 2x - 1}{\cos 2x + 1} dx =\]
\[\int\frac{x + 2}{\left( x + 1 \right)^3} \text{ dx }\]
\[\int\frac{\sin x}{\sqrt{1 + \sin x}} dx\]
\[\int \sin^3 x \cos^4 x\ \text{ dx }\]
\[\int\sqrt{\sin x} \cos^3 x\ \text{ dx }\]
\[\int\sqrt{\frac{1 + x}{x}} \text{ dx }\]
\[\int\frac{1}{1 + 2 \cos x} \text{ dx }\]
\[\int\frac{1}{\sin^4 x + \cos^4 x} \text{ dx}\]
\[\int\frac{\sin^2 x}{\cos^6 x} \text{ dx }\]
\[\int\left( 2x + 3 \right) \sqrt{4 x^2 + 5x + 6} \text{ dx}\]
\[\int x \sec^2 2x\ dx\]
\[\int\frac{x^2}{\left( x - 1 \right)^3 \left( x + 1 \right)} \text{ dx}\]
\[\int\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} \text{ dx}\]
