Advertisements
Advertisements
प्रश्न
\[\int\frac{x^6 + 1}{x^2 + 1} dx\]
योग
उत्तर
\[\int \left( \frac{x^6 + 1}{x^2 + 1} \right)dx\]
\[ = \int \left[ \frac{\left( x^2 \right)^3 + 1^3}{x^2 + 1} \right]\text{dx }A^3 + B^3 = \left( A + B \right) \left( A^2 - AB + B^2 \right)\]
\[ = \int\frac{\left( x^2 + 1 \right)\left( x^4 - x^2 + 1 \right)}{\left( x^2 + 1 \right)}dx\]
\[ = \int\left( x^4 - x^2 + 1 \right)dx\]
\[ = \int x^4 dx + \int x^2 dx + \int1dx\]
\[ = \frac{x^{4 + 1}}{4 + 1} - \frac{x^{2 + 1}}{2 + 1} + x + C\]
\[ = \frac{x^5}{5} - \frac{x^3}{3} + x + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx\]
\[\int \left( \tan x + \cot x \right)^2 dx\]
\[\int\frac{1}{1 - \cos 2x} dx\]
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
\[\int\frac{1}{2 - 3x} + \frac{1}{\sqrt{3x - 2}} dx\]
\[\int\frac{x + 3}{\left( x + 1 \right)^4} dx\]
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
\[\int\left( x + 2 \right) \sqrt{3x + 5} \text{dx} \]
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
\[\int\frac{a}{b + c e^x} dx\]
\[\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x} dx\]
\[\int\frac{\cos^3 x}{\sqrt{\sin x}} dx\]
\[\int 5^{x + \tan^{- 1} x} . \left( \frac{x^2 + 2}{x^2 + 1} \right) dx\]
\[\int\frac{1}{\sqrt{x} + x} \text{ dx }\]
\[\int\frac{1}{x\sqrt{4 - 9 \left( \log x \right)^2}} dx\]
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} dx\]
\[\int\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int\frac{1}{2 + \sin x + \cos x} \text{ dx }\]
\[\int\frac{2 \sin x + 3 \cos x}{3 \sin x + 4 \cos x} dx\]
\[\int x^2 \text{ cos x dx }\]
` ∫ x tan ^2 x dx
\[\int\left( e^\text{log x} + \sin x \right) \text{ cos x dx }\]
\[\int e^x \left( \log x + \frac{1}{x^2} \right) dx\]
\[\int x^2 \sqrt{a^6 - x^6} \text{ dx}\]
\[\int\frac{5 x^2 - 1}{x \left( x - 1 \right) \left( x + 1 \right)} dx\]
\[\int\frac{\cos x}{\left( 1 - \sin x \right)^3 \left( 2 + \sin x \right)} dx\]
\[\int\frac{x^2 - 1}{x^4 + 1} \text{ dx }\]
If \[\int\frac{1}{5 + 4 \sin x} dx = A \tan^{- 1} \left( B \tan\frac{x}{2} + \frac{4}{3} \right) + C,\] then
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{1}{e^x + 1} \text{ dx }\]
\[\int\frac{1}{4 x^2 + 4x + 5} dx\]
\[\int\frac{x + 1}{x^2 + 4x + 5} \text{ dx}\]
\[\int\sqrt{x^2 - a^2} \text{ dx}\]
\[\int \left( x + 1 \right)^2 e^x \text{ dx }\]
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx}\]
\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]
\[\int\frac{x^2 + 1}{x^2 - 5x + 6} \text{ dx }\]
