Advertisements
Advertisements
प्रश्न
\[\int\frac{1}{x^{2/3} \sqrt{x^{2/3} - 4}} dx\]
योग
Advertisements
उत्तर
\[ = \int\frac{dx}{x^\frac{2}{3} \sqrt{\left( x^\frac{1}{3} \right)^2 - 2^2}}\]
\[ = \int\frac{dx}{x^\frac{2}{3} \sqrt{\left( x^\frac{1}{3} \right)^2 - 2^2}}\]
\[\text{ Let } x^\frac{1}{3} = t\]
\[ \Rightarrow \frac{1}{3} x^\frac{- 2}{3} dx = dt\]
\[ \Rightarrow \frac{1}{3 x^\frac{2}{3}} dx = dt\]
\[ \Rightarrow \frac{dx}{x^\frac{2}{3}} = 3 dt\]
\[Now, \int\frac{dx}{x^\frac{2}{3} \sqrt{x^\frac{2}{3} - 2^2}}\]
\[ = 3\int\frac{dt}{\sqrt{t^2 - 2^2}}\]
\[ = 3 \text{ log } \left| t + \sqrt{t^2 - 2^2} \right| + C\]
\[ = 3 \text{ log }\left| x^\frac{1}{3} + \sqrt{x^\frac{2}{3} - 4} \right| + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
`int{sqrtx(ax^2+bx+c)}dx`
\[\int\sqrt{x}\left( x^3 - \frac{2}{x} \right) dx\]
\[\int\frac{x^5 + x^{- 2} + 2}{x^2} dx\]
\[\int\frac{1}{1 + \cos 2x} dx\]
\[\int \sin^{- 1} \left( \frac{2 \tan x}{1 + \tan^2 x} \right) dx\]
\[\int \left( a \tan x + b \cot x \right)^2 dx\]
\[\int\frac{1}{\sqrt{x + 1} + \sqrt{x}} dx\]
\[\int \sin^2 \frac{x}{2} dx\]
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
\[\int\frac{1}{x (3 + \log x)} dx\]
\[\int\sqrt {e^x- 1} \text{dx}\]
\[\int\frac{1}{\sin x \cos^3 x} dx\]
\[\int\frac{1}{a^2 x^2 + b^2} dx\]
\[\int\frac{\sec^2 x}{1 - \tan^2 x} dx\]
\[\int\frac{2x - 3}{x^2 + 6x + 13} dx\]
\[\int\frac{2x + 1}{\sqrt{x^2 + 2x - 1}}\text{ dx }\]
\[\int\frac{1}{\left( \sin x - 2 \cos x \right)\left( 2 \sin x + \cos x \right)} \text{ dx }\]
\[\int\frac{1}{1 - \cot x} dx\]
\[\int\frac{3 + 2 \cos x + 4 \sin x}{2 \sin x + \cos x + 3} \text{ dx }\]
\[\int\frac{4 \sin x + 5 \cos x}{5 \sin x + 4 \cos x} \text{ dx }\]
` ∫ sin x log (\text{ cos x ) } dx `
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx }\]
\[\int\frac{e^x \left( x - 4 \right)}{\left( x - 2 \right)^3} \text{ dx }\]
\[\int\sqrt{3 - 2x - 2 x^2} \text{ dx}\]
\[\int\frac{1}{x\left( x - 2 \right) \left( x - 4 \right)} dx\]
\[\int\frac{3 + 4x - x^2}{\left( x + 2 \right) \left( x - 1 \right)} dx\]
\[\int\frac{5x}{\left( x + 1 \right) \left( x^2 - 4 \right)} dx\]
\[\int\frac{5 x^2 - 1}{x \left( x - 1 \right) \left( x + 1 \right)} dx\]
\[\int\frac{x^2 + 1}{\left( 2x + 1 \right) \left( x^2 - 1 \right)} dx\]
\[\int\frac{1}{1 + x + x^2 + x^3} dx\]
Find \[\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)^2}dx\]
\[\int\frac{\left( x^2 + 1 \right) \left( x^2 + 2 \right)}{\left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx\]
Evaluate the following integral:
\[\int\frac{x^2}{1 - x^4}dx\]
\[\int\frac{\cos 2x - 1}{\cos 2x + 1} dx =\]
\[\int\frac{x^3}{\left( 1 + x^2 \right)^2} \text{ dx }\]
\[\int\sqrt{\sin x} \cos^3 x\ \text{ dx }\]
\[\int\frac{1 + \sin x}{\sin x \left( 1 + \cos x \right)} \text{ dx }\]
\[\int\frac{1 + x^2}{\sqrt{1 - x^2}} \text{ dx }\]
