Advertisements
Advertisements
Question
\[\int\frac{1}{\sqrt{x}}\left( 1 + \frac{1}{x} \right) dx\]
Sum
Advertisements
Solution
\[\int\frac{1}{\sqrt{x}}\left( 1 + \frac{1}{x} \right)dx\]
\[ = \int x^{- \frac{1}{2}} \left( 1 + \frac{1}{x} \right)dx\]
\[ = \int\left( x^{- \frac{1}{2}} + \frac{1}{x^\frac{3}{2}} \right)dx\]
\[ = \int x^{- \frac{1}{2}} dx + \int x^{- \frac{3}{2}} dx\]
\[ = \left[ \frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} \right] + \left[ \frac{x^{- \frac{3}{2} + 1}}{- \frac{3}{2} + 1} \right]\]
\[ = 2\sqrt{x} - \frac{2}{\sqrt{x}} + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx\]
\[\int\frac{1}{1 - \sin x} dx\]
\[\int \left( 2x - 3 \right)^5 + \sqrt{3x + 2} \text{dx} \]
\[\int\frac{1 + \cos x}{1 - \cos x} dx\]
\[\int\frac{1}{1 - \sin\frac{x}{2}} dx\]
\[\int\frac{\sin 2x}{\sin 5x \sin 3x} dx\]
\[\int\frac{\cos x - \sin x}{1 + \sin 2x} dx\]
\[\int\frac{\sin\sqrt{x}}{\sqrt{x}} dx\]
\[\int \tan^3 \text{2x sec 2x dx}\]
\[\int\left( 2 x^2 + 3 \right) \sqrt{x + 2} \text{ dx }\]
\[\int\frac{x^2}{\sqrt{1 - x}} dx\]
\[\int \sin^3 x \cos^5 x \text{ dx }\]
\[\int\frac{1}{\sin^3 x \cos^5 x} dx\]
\[\int\frac{x}{x^4 + 2 x^2 + 3} dx\]
\[\int\frac{1}{x\sqrt{4 - 9 \left( \log x \right)^2}} dx\]
\[\int\frac{\left( 3 \sin x - 2 \right) \cos x}{5 - \cos^2 x - 4 \sin x} dx\]
\[\int\frac{\left( x - 1 \right)^2}{x^2 + 2x + 2} dx\]
\[\int\frac{1}{\cos x \left( \sin x + 2 \cos x \right)} dx\]
\[\int\frac{3 + 2 \cos x + 4 \sin x}{2 \sin x + \cos x + 3} \text{ dx }\]
\[\int x \cos^2 x\ dx\]
\[\int x^3 \cos x^2 dx\]
\[\int\frac{x + \sin x}{1 + \cos x} \text{ dx }\]
\[\int\left( x + 1 \right) \text{ log x dx }\]
\[\int e^x \left( \log x + \frac{1}{x} \right) dx\]
\[\int x\sqrt{x^2 + x} \text{ dx }\]
\[\int\frac{3 + 4x - x^2}{\left( x + 2 \right) \left( x - 1 \right)} dx\]
\[\int\frac{1}{\sin x \left( 3 + 2 \cos x \right)} dx\]
\[\int\frac{1}{\left( 1 + x^2 \right) \sqrt{1 - x^2}} \text{ dx }\]
\[\int e^x \left( 1 - \cot x + \cot^2 x \right) dx =\]
\[\int\frac{1}{1 - \cos x - \sin x} dx =\]
The primitive of the function \[f\left( x \right) = \left( 1 - \frac{1}{x^2} \right) a^{x + \frac{1}{x}} , a > 0\text{ is}\]
\[\int\frac{1}{\sqrt{x} + \sqrt{x + 1}} \text{ dx }\]
\[\int\sqrt{\sin x} \cos^3 x\ \text{ dx }\]
\[\int\frac{1}{\sqrt{x^2 - a^2}} \text{ dx }\]
\[\int \left( \sin^{- 1} x \right)^3 dx\]
\[\int\frac{e^{m \tan^{- 1} x}}{\left( 1 + x^2 \right)^{3/2}} \text{ dx}\]
\[\int \left( e^x + 1 \right)^2 e^x dx\]
\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]
