Advertisements
Advertisements
प्रश्न
` ∫ x {tan^{- 1} x^2}/{1 + x^4} dx`
योग
Advertisements
उत्तर
\[\int\frac{x \tan^{- 1} x^2}{1 + x^4} dx\]
\[\text{Let} \tan^{- 1} x^2 = t\]
\[ \Rightarrow \frac{1}{1 + \left( x^2 \right)^2} \times 2x = \frac{dt}{dx}\]
` ⇒ {x dx}/{1 + x^4} = {dt}/{2}`
\[Now, \int\frac{x \tan^{- 1} x^2}{1 + x^4} dx\]
\[ = \frac{1}{2}\ ∫ t . dt\]
\[ = \frac{1}{2} \times \frac{t^2}{2} + C\]
\[ = \frac{\left( \tan^{- 1} x^2 \right)^2}{4} + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\]
\[\int\frac{\sin^3 x - \cos^3 x}{\sin^2 x \cos^2 x} dx\]
\[\int\frac{5 \cos^3 x + 6 \sin^3 x}{2 \sin^2 x \cos^2 x} dx\]
\[\int\frac{1}{\sqrt{x + a} + \sqrt{x + b}} dx\]
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
\[\int \cos^2 \text{nx dx}\]
Integrate the following integrals:
\[\int\text{sin 2x sin 4x sin 6x dx} \]
` = ∫ root (3){ cos^2 x} sin x dx `
\[\int\frac{x \sin^{- 1} x^2}{\sqrt{1 - x^4}} dx\]
\[\int\frac{x^5}{\sqrt{1 + x^3}} dx\]
\[\int\frac{x^2}{\sqrt{1 - x}} dx\]
\[\int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx\]
` ∫ sec^6 x tan x dx `
\[\int\frac{1}{x^2 - 10x + 34} dx\]
\[\int\frac{1}{x^{2/3} \sqrt{x^{2/3} - 4}} dx\]
\[\int\frac{\cos x}{\sqrt{\sin^2 x - 2 \sin x - 3}} dx\]
\[\int\frac{x}{x^2 + 3x + 2} dx\]
\[\int\frac{2x + 5}{\sqrt{x^2 + 2x + 5}} dx\]
\[\int\frac{5x + 3}{\sqrt{x^2 + 4x + 10}} \text{ dx }\]
\[\int\frac{1}{1 + 3 \sin^2 x} \text{ dx }\]
\[\int\frac{1}{5 - 4 \sin x} \text{ dx }\]
\[\int\frac{1}{4 \cos x - 1} \text{ dx }\]
\[\int\frac{\log \left( \log x \right)}{x} dx\]
\[\int \sec^{- 1} \sqrt{x}\ dx\]
\[\int e^x \left( \cot x - {cosec}^2 x \right) dx\]
∴\[\int e^{2x} \left( - \sin x + 2 \cos x \right) dx\]
\[\int\frac{1}{\cos x + \sqrt{3} \sin x} \text{ dx } \] is equal to
\[\int e^x \left( 1 - \cot x + \cot^2 x \right) dx =\]
If \[\int\frac{1}{\left( x + 2 \right)\left( x^2 + 1 \right)}dx = a\log\left| 1 + x^2 \right| + b \tan^{- 1} x + \frac{1}{5}\log\left| x + 2 \right| + C,\] then
\[\int\frac{1}{\sqrt{x} + \sqrt{x + 1}} \text{ dx }\]
\[\int\frac{1}{e^x + 1} \text{ dx }\]
\[\int\frac{1}{e^x + e^{- x}} dx\]
\[\int\frac{\sin x}{\sqrt{1 + \sin x}} dx\]
\[\int\frac{\sqrt{a} - \sqrt{x}}{1 - \sqrt{ax}}\text{ dx }\]
\[\int\frac{1}{1 + 2 \cos x} \text{ dx }\]
\[\int \sec^6 x\ dx\]
\[\int\sqrt{a^2 + x^2} \text{ dx }\]
\[\int\frac{x^5}{\sqrt{1 + x^3}} \text{ dx }\]
\[\int\frac{\sin x + \cos x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{x^2}{x^2 + 7x + 10}\text{ dx }\]
