Advertisements
Advertisements
प्रश्न
\[\int e^\sqrt{x} \text{ dx }\]
बेरीज
Advertisements
उत्तर
\[\text{ Let I } = \int e^\sqrt{x} \text{ dx }\]
\[ = \int\sqrt{x} \cdot \frac{e^\sqrt{x}}{\sqrt{x}}dx\]
\[\text{ Let }\sqrt{x} = t\]
\[ \Rightarrow \frac{1}{2\sqrt{x}}dx = dt\]
\[ \Rightarrow \frac{dx}{\sqrt{x}} = 2 dt\]
\[ \therefore I = 2\int t_{} \cdot {e^t}_{} dt\]
` " Taking t as the first function and e"^t" as the second function " . `
\[ = 2\left[ t\int e^t dt - \int\left\{ \frac{d}{dt}\left( t \right)\int e^t dt \right\}dt \right] \]
\[ = 2\left[ t \cdot e^t - \int1 \cdot e^t dt \right] + C . . . (1)\]
\[\text{Substituting the value of t in eq} \text{ (1) }\]
\[ = 2\left[ \sqrt{x} \text{ e }^\sqrt{x} - e^\sqrt{x} \right] + C\]
\[ = 2 \text{ e}^\sqrt{x} \left( \sqrt{x} - 1 \right) + C\]
shaalaa.com
या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{1}{1 - \cos x} dx\]
\[\int\frac{\tan x}{\sec x + \tan x} dx\]
\[\int\frac{1}{1 + \cos 2x} dx\]
\[\int\frac{1}{\sqrt{2x + 3} + \sqrt{2x - 3}} dx\]
\[\int\frac{1}{\sqrt{1 + \cos x}} dx\]
\[\int\frac{\text{sin} \left( x - \alpha \right)}{\text{sin }\left( x + \alpha \right)} dx\]
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
\[\int\frac{1}{x (3 + \log x)} dx\]
\[\int\frac{1 - \sin 2x}{x + \cos^2 x} dx\]
\[\int\frac{1}{1 + \sqrt{x}} dx\]
\[\int\frac{x \sin^{- 1} x^2}{\sqrt{1 - x^4}} dx\]
` = ∫1/{sin^3 x cos^ 2x} dx`
\[\int\frac{1}{\sin x \cos^3 x} dx\]
\[\int\frac{x^2 - 1}{x^2 + 4} dx\]
\[\int\frac{\cos x}{\sqrt{4 + \sin^2 x}} dx\]
\[\int\frac{x + 1}{x^2 + x + 3} dx\]
\[\int\frac{\left( x - 1 \right)^2}{x^2 + 2x + 2} dx\]
\[\int\frac{x}{\sqrt{x^2 + x + 1}} \text{ dx }\]
\[\int\frac{1}{\sin x + \sqrt{3} \cos x} \text{ dx }\]
\[\int\frac{2 \tan x + 3}{3 \tan x + 4} \text{ dx }\]
\[\int\frac{\left( x \tan^{- 1} x \right)}{\left( 1 + x^2 \right)^{3/2}} \text{ dx }\]
\[\int x^3 \tan^{- 1}\text{ x dx }\]
\[\int \cos^3 \sqrt{x}\ dx\]
\[\int e^x \left( \cos x - \sin x \right) dx\]
\[\int\sqrt{3 - 2x - 2 x^2} \text{ dx}\]
\[\int\sqrt{x^2 - 2x} \text{ dx}\]
\[\int\frac{2 x^2 + 7x - 3}{x^2 \left( 2x + 1 \right)} dx\]
\[\int\frac{3x + 5}{x^3 - x^2 - x + 1} dx\]
\[\int\frac{x + 1}{x \left( 1 + x e^x \right)} dx\]
\[\int\frac{x^2 - 1}{x^4 + 1} \text{ dx }\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }\]
\[\int\frac{1}{\left( 2 x^2 + 3 \right) \sqrt{x^2 - 4}} \text{ dx }\]
\[\int\frac{\cos2x - \cos2\theta}{\cos x - \cos\theta}dx\] is equal to
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{1}{\left( \sin x - 2 \cos x \right) \left( 2 \sin x + \cos x \right)} \text{ dx }\]
\[\int\log \left( x + \sqrt{x^2 + a^2} \right) \text{ dx}\]
\[\int\frac{1 + x^2}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int x\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx}\]
\[\int\frac{e^{m \tan^{- 1} x}}{\left( 1 + x^2 \right)^{3/2}} \text{ dx}\]
