Advertisements
Advertisements
प्रश्न
\[\int x^2 \sqrt{a^6 - x^6} \text{ dx}\]
योग
Advertisements
उत्तर
\[\text{ Let I } = \int x^2 \sqrt{a^6 - x^6}\text{ \text{ dx}}\]
\[ = \int x^2 \sqrt{\left( a^3 \right)^2 - \left( x^3 \right)^2}\text{ dx}\]
\[Putting\ x^3 = t\]
\[ \Rightarrow 3 x^2 dx = dt\]
\[ \Rightarrow x^2 dx = \frac{dt}{3}\]
\[ \therefore I = \frac{1}{3}\int\sqrt{\left( a^3 \right)^2 - t^2}dt\]
\[ = \frac{1}{3} \left[ \frac{t}{2}\sqrt{\left( a^3 \right)^2 - t^2} + \frac{\left( a^3 \right)^2}{2} \text{ sin}^{- 1} \left( \frac{t}{a^3} \right) \right] + C\]
\[ = \frac{x^3}{6} \sqrt{a^6 - x^6} + \frac{a^6}{6} \text{ sin}^{- 1} \left( \frac{x^3}{a^3} \right) + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx\]
\[\int \cos^{- 1} \left( \sin x \right) dx\]
\[\int\frac{\cos x}{1 + \cos x} dx\]
\[\int\frac{1}{\text{cos}^2\text{ x }\left( 1 - \text{tan x} \right)^2} dx\]
\[\int\frac{2x + 3}{\left( x - 1 \right)^2} dx\]
\[\int\frac{2 - 3x}{\sqrt{1 + 3x}} dx\]
\[\int\frac{\cos x}{2 + 3 \sin x} dx\]
` = ∫ root (3){ cos^2 x} sin x dx `
\[\int \sin^5\text{ x }\text{cos x dx}\]
\[\int\frac{1}{a^2 x^2 + b^2} dx\]
\[\int\frac{1}{\sqrt{a^2 + b^2 x^2}} dx\]
\[\int\frac{1}{\sqrt{a^2 - b^2 x^2}} dx\]
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 - 1}} dx\]
\[\int\frac{1}{\sqrt{5 x^2 - 2x}} dx\]
\[\int\frac{\sec^2 x}{\sqrt{4 + \tan^2 x}} dx\]
` ∫ {x-3} /{ x^2 + 2x - 4 } dx `
\[\int\frac{x^3}{x^4 + x^2 + 1}dx\]
\[\int\frac{x}{\sqrt{x^2 + 6x + 10}} \text{ dx }\]
\[\int\frac{x + 2}{\sqrt{x^2 + 2x - 1}} \text{ dx }\]
\[\int\frac{1}{3 + 2 \cos^2 x} \text{ dx }\]
\[\int \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ dx }\]
\[\int e^x \sec x \left( 1 + \tan x \right) dx\]
\[\int e^x \left( \cot x + \log \sin x \right) dx\]
\[\int\frac{\sqrt{1 - \sin x}}{1 + \cos x} e^{- x/2} \text{ dx }\]
\[\int e^x \left( \frac{\sin x \cos x - 1}{\sin^2 x} \right) dx\]
\[\int\frac{1}{x\left( x - 2 \right) \left( x - 4 \right)} dx\]
\[\int\frac{x^2 + 1}{x\left( x^2 - 1 \right)} dx\]
\[\int\frac{1}{\cos x \left( 5 - 4 \sin x \right)} dx\]
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} dx\]
\[\int\frac{1}{\left( x + 1 \right) \sqrt{x^2 + x + 1}} \text{ dx }\]
\[\int\frac{1}{1 - \cos x - \sin x} dx =\]
\[\int e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx\]
\[\int\frac{1}{\sqrt{x} + \sqrt{x + 1}} \text{ dx }\]
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} \text{ dx }\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{1}{\sqrt{x^2 - a^2}} \text{ dx }\]
\[\int \tan^5 x\ \sec^3 x\ dx\]
\[\int\sqrt{1 + 2x - 3 x^2}\text{ dx } \]
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx}\]
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
