Advertisements
Advertisements
Question
\[\int\frac{1}{1 + \sqrt{x}} dx\]
Sum
Solution
\[\int \frac{dx}{1 + \sqrt{x}}\]
\[ = \int\frac{\sqrt{x} dx}{\sqrt{x} \left( 1 + \sqrt{x} \right)}\]
\[\text{Let 1} + \sqrt{x} = t\]
\[ \Rightarrow \sqrt{x} = t - 1\]
\[ \Rightarrow \frac{1}{2\sqrt{x}} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{dx}{\sqrt{x}} = 2dt\]
\[Now, \int\frac{\sqrt{x}}{\sqrt{x}\left( 1 + \sqrt{x} \right)}dx\]
\[ = 2\int\left( \frac{t - 1}{t} \right)dt\]
\[ = 2\int\left( 1 - \frac{1}{t} \right)dt\]
\[ = 2 \left( t - \text{log} \left| t \right| \right) + C\]
\[ = 2 \left( 1 + \sqrt{x} \right) - 2 \log \left| 1 + \sqrt{x} \right| + C\]
\[\text{Let} \text{ C }+ 2 = C'\]
\[ = 2\sqrt{x} - \text{2 log} \left( 1 + \sqrt{x} \right) + C'\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{x^6 + 1}{x^2 + 1} dx\]
\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]
\[\int \cos^2 \frac{x}{2} dx\]
\[\int \cos^2 \text{nx dx}\]
\[\int\frac{\text{sin} \left( x - \alpha \right)}{\text{sin }\left( x + \alpha \right)} dx\]
\[\int\frac{1}{ x \text{log x } \text{log }\left( \text{log x }\right)} dx\]
\[\int\frac{\cos^3 x}{\sqrt{\sin x}} dx\]
\[\int\frac{e^\sqrt{x} \cos \left( e^\sqrt{x} \right)}{\sqrt{x}} dx\]
\[\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} dx\]
\[\int 5^{5^{5^x}} 5^{5^x} 5^x dx\]
\[\int x^2 \sqrt{x + 2} \text{ dx }\]
\[\int\frac{1}{\sqrt{3 x^2 + 5x + 7}} dx\]
\[\int\frac{\cos 2x}{\sqrt{\sin^2 2x + 8}} dx\]
\[\int\frac{1}{x^{2/3} \sqrt{x^{2/3} - 4}} dx\]
\[\int\frac{2x}{2 + x - x^2} \text{ dx }\]
\[\int\frac{2x + 1}{\sqrt{x^2 + 2x - 1}}\text{ dx }\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{1}{3 + 2 \cos^2 x} \text{ dx }\]
\[\int\frac{1}{13 + 3 \cos x + 4 \sin x} dx\]
\[\int\frac{1}{5 + 7 \cos x + \sin x} dx\]
\[\int x^2 e^{- x} \text{ dx }\]
\[\int\cos\sqrt{x}\ dx\]
\[\int \sin^{- 1} \sqrt{x} \text{ dx }\]
\[\int \sin^3 \sqrt{x}\ dx\]
\[\int e^x \left[ \sec x + \log \left( \sec x + \tan x \right) \right] dx\]
\[\int e^x \left( \log x + \frac{1}{x} \right) dx\]
\[\int\sqrt{3 - 2x - 2 x^2} \text{ dx}\]
\[\int\frac{x^2 + 1}{x^2 - 1} dx\]
\[\int\frac{x^3}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{x}{\left( x - 1 \right)^2 \left( x + 2 \right)} dx\]
\[\int\frac{x^2 - 1}{x^4 + 1} \text{ dx }\]
\[\int\frac{x^9}{\left( 4 x^2 + 1 \right)^6}dx\] is equal to
\[\int\frac{1 - x^4}{1 - x} \text{ dx }\]
\[\int\frac{\sin x}{1 + \sin x} \text{ dx }\]
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} \text{ dx }\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\sqrt{3 x^2 + 4x + 1}\text{ dx }\]
\[\int\log \left( x + \sqrt{x^2 + a^2} \right) \text{ dx}\]
\[\int x^3 \left( \log x \right)^2\text{ dx }\]
\[\int\frac{\sin x + \cos x}{\sin^4 x + \cos^4 x} \text{ dx }\]
