Advertisements
Advertisements
प्रश्न
\[\int\frac{x}{\sqrt{x^4 + a^4}} dx\]
योग
Advertisements
उत्तर
` ∫ {x dx}/{\sqrt{x^4 + a^4}} `
` ∫ {x dx}/\sqrt{(x^2)^2 + (a^2)^2}`
` \text{ let} x^2 = t `
\[ \Rightarrow\text{ 2x dx } = dt\]
\[ \Rightarrow\text{ x dx } = \frac{dt}{2}\]
Now, ` ∫ {x dx}/\sqrt{(x^2)^2 + (a^2)^2}`
\[ = \frac{1}{2}\int\frac{dt}{\sqrt{t^2 + \left( a^2 \right)^2}}\]
\[ = \frac{1}{2} \text{ log }\left| t + \sqrt{t^2 + a^4} \right| + C\]
\[ = \frac{1}{2} \text{ log }\left| x^2 + \sqrt{x^4 + a^4} \right| + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\left( 3x\sqrt{x} + 4\sqrt{x} + 5 \right)dx\]
\[\int\frac{x^3 - 3 x^2 + 5x - 7 + x^2 a^x}{2 x^2} dx\]
\[\int\frac{1}{\sqrt{x + 1} + \sqrt{x}} dx\]
\[\int\sin x\sqrt{1 + \cos 2x} dx\]
\[\int\sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}} dx\]
\[\int\left\{ 1 + \tan x \tan \left( x + \theta \right) \right\} dx\]
\[\int\frac{x}{\sqrt{x^2 + a^2} + \sqrt{x^2 - a^2}} dx\]
\[\int \sin^3 x \cos^6 x \text{ dx }\]
\[\int \sin^7 x \text{ dx }\]
\[\int\frac{1}{\sqrt{7 - 6x - x^2}} dx\]
\[\int\frac{1}{\sqrt{5 x^2 - 2x}} dx\]
\[\int\frac{x}{\sqrt{4 - x^4}} dx\]
\[\int\frac{x^2 + x + 1}{x^2 - x + 1} \text{ dx }\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
`int 1/(sin x - sqrt3 cos x) dx`
\[\int x^3 \text{ log x dx }\]
\[\int \left( \log x \right)^2 \cdot x\ dx\]
\[\int\left( \tan^{- 1} x^2 \right) x\ dx\]
\[\int e^x \left( \log x + \frac{1}{x^2} \right) dx\]
\[\int\left( x + 1 \right) \sqrt{x^2 + x + 1} \text{ dx }\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{2x - 3}{\left( x^2 - 1 \right) \left( 2x + 3 \right)} dx\]
\[\int\frac{\sin 2x}{\left( 1 + \sin x \right) \left( 2 + \sin x \right)} dx\]
\[\int\frac{1}{x\left( x^n + 1 \right)} dx\]
\[\int\frac{1}{x \left( x^4 + 1 \right)} dx\]
\[\int\frac{1}{\cos x \left( 5 - 4 \sin x \right)} dx\]
\[\int\frac{1}{\sin x + \sin 2x} dx\]
\[\int\frac{\left( x^2 + 1 \right) \left( x^2 + 2 \right)}{\left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx\]
\[\int\frac{1}{x^4 + x^2 + 1} \text{ dx }\]
If \[\int\frac{\sin^8 x - \cos^8 x}{1 - 2 \sin^2 x \cos^2 x} dx\]
\[\int\frac{1 - x^4}{1 - x} \text{ dx }\]
\[\int\sin x \sin 2x \text{ sin 3x dx }\]
\[\int\frac{1}{\sin x + \sin 2x} \text{ dx }\]
\[\int\frac{\sin^5 x}{\cos^4 x} \text{ dx }\]
\[\int\frac{\log \left( \log x \right)}{x} \text{ dx}\]
\[\int\frac{1}{x\sqrt{1 + x^3}} \text{ dx}\]
\[\int \cos^{- 1} \left( 1 - 2 x^2 \right) \text{ dx }\]
\[\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} \text{ dx}\]
\[\int\frac{\cos^7 x}{\sin x} dx\]
