Advertisements
Advertisements
प्रश्न
\[\int\frac{e^{2x}}{1 + e^x} dx\]
योग
Advertisements
उत्तर
\[\int\frac{e^{2x} dx}{1 + e^x}\]
\[ \Rightarrow \int\frac{e^x . e^x}{1 + e^x}dx\]
\[\text{Let 1 }+ e^x = t \]
\[ \Rightarrow e^x = t - 1\]
\[ \Rightarrow e^x dx = dt\]
\[Now, \int\frac{e^x . e^x}{1 + e^x}dx\]
\[ = \int \frac{\left( t - 1 \right) . dt}{t}\]
\[ = \left( 1 - \frac{1}{t} \right)dt\]
\[ = t - \text{log }\left| t \right| + C\]
\[ = \left( 1 + e^x \right) - \log \left( 1 + e^x \right) + C\]
\[\text{Let C }+ 1 = C'\]
\[ = e^x - \text{log} \left( \text{1 + e}^x \right) + C'\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx\]
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\]
\[\int\sqrt{x}\left( 3 - 5x \right) dx\]
\[\int\frac{1}{\sqrt{1 - \cos 2x}} dx\]
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
\[\int\frac{\cos^5 x}{\sin x} dx\]
` ∫ x {tan^{- 1} x^2}/{1 + x^4} dx`
\[\int\frac{x^2 + 3x + 1}{\left( x + 1 \right)^2} dx\]
` ∫ tan x sec^4 x dx `
\[\int \sin^3 x \cos^6 x \text{ dx }\]
\[\int\frac{e^x}{1 + e^{2x}} dx\]
\[\int\frac{1}{\sqrt{2x - x^2}} dx\]
\[\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)\]
\[\int\frac{e^x}{\sqrt{16 - e^{2x}}} dx\]
\[\int\frac{\left( 3 \sin x - 2 \right) \cos x}{5 - \cos^2 x - 4 \sin x} dx\]
\[\int\frac{\left( x - 1 \right)^2}{x^2 + 2x + 2} dx\]
\[\int\frac{2x + 1}{\sqrt{x^2 + 4x + 3}} \text{ dx }\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{1}{\cos 2x + 3 \sin^2 x} dx\]
\[\int\frac{1}{5 + 4 \cos x} dx\]
\[\int \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) \text{ dx }\]
\[\int\left( \tan^{- 1} x^2 \right) x\ dx\]
\[\int e^x \left( \frac{\sin x \cos x - 1}{\sin^2 x} \right) dx\]
\[\int(2x + 5)\sqrt{10 - 4x - 3 x^2}dx\]
\[\int\frac{2 x^2 + 7x - 3}{x^2 \left( 2x + 1 \right)} dx\]
\[\int\frac{x}{\left( x + 1 \right) \left( x^2 + 1 \right)} dx\]
\[\int\frac{1}{x^4 - 1} dx\]
\[\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 9}} \text{ dx}\]
\[\int\text{ cos x cos 2x cos 3x dx}\]
\[\int\frac{1}{\text{ cos }\left( x - a \right) \text{ cos }\left( x - b \right)} \text{ dx }\]
\[\int\frac{x^3}{\left( 1 + x^2 \right)^2} \text{ dx }\]
\[\int\sqrt{\text{ cosec x} - 1} \text{ dx }\]
\[\int\frac{1}{a + b \tan x} \text{ dx }\]
\[\int\frac{1}{\sin^4 x + \cos^4 x} \text{ dx}\]
\[\int\sqrt{x^2 - a^2} \text{ dx}\]
\[\int\sqrt{3 x^2 + 4x + 1}\text{ dx }\]
\[\int\frac{1}{x \sqrt{1 + x^n}} \text{ dx}\]
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
\[\int\frac{x^2}{x^2 + 7x + 10}\text{ dx }\]
