Advertisements
Advertisements
Question
\[\int\frac{x^2}{x^6 + a^6} dx\]
Sum
Solution
\[\int\frac{x^2}{x^6 + a^6}dx\]
\[ \Rightarrow \int\frac{x^2 dx}{\left( x^3 \right)^2 + \left( a^3 \right)^2}\]
\[\text{ let } x^3 = t\]
\[ \Rightarrow 3 x^2 dx = dt\]
\[ \Rightarrow x^2 dx = \frac{dt}{3}\]
\[Now, \int\frac{x^2}{x^6 + a^6}dx\]
\[ = \frac{1}{3}\int\frac{dt}{t^2 + \left( a^3 \right)^2}\]
\[ = \frac{1}{3 a^3} \tan^{- 1} \left( \frac{t}{a^3} \right) + C\]
\[ = \frac{1}{3 a^3} \tan-^1 \left( \frac{x^3}{a^3} \right) + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx\]
\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]
\[\int\frac{1}{1 - \sin x} dx\]
Write the primitive or anti-derivative of
\[f\left( x \right) = \sqrt{x} + \frac{1}{\sqrt{x}} .\]
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
\[\int\frac{x^2 + 5x + 2}{x + 2} dx\]
\[\int\frac{2x + 1}{\sqrt{3x + 2}} dx\]
\[\int\frac{x}{\sqrt{x + a} - \sqrt{x + b}}dx\]
\[\int\frac{e^x + 1}{e^x + x} dx\]
\[\int\frac{1}{ x \text{log x } \text{log }\left( \text{log x }\right)} dx\]
\[\int\frac{2 \cos 2x + \sec^2 x}{\sin 2x + \tan x - 5} dx\]
\[\int\frac{\sin 2x}{\sin 5x \sin 3x} dx\]
` = ∫ root (3){ cos^2 x} sin x dx `
\[\int\frac{\sin 2x}{\left( a + b \cos 2x \right)^2} dx\]
\[\int\frac{\left( x + 1 \right) e^x}{\sin^2 \left( \text{x e}^x \right)} dx\]
\[\int\frac{1}{\left( x + 1 \right)\left( x^2 + 2x + 2 \right)} dx\]
\[\int\frac{1}{\sqrt{x} + x} \text{ dx }\]
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
\[\int\left( 2 x^2 + 3 \right) \sqrt{x + 2} \text{ dx }\]
` ∫ tan^5 x dx `
` ∫ \sqrt{tan x} sec^4 x dx `
\[\int \sin^4 x \cos^3 x \text{ dx }\]
\[\int\frac{\sec^2 x}{\sqrt{4 + \tan^2 x}} dx\]
\[\int\frac{x}{\sqrt{x^2 + x + 1}} \text{ dx }\]
\[\int\frac{2}{2 + \sin 2x}\text{ dx }\]
\[\int\frac{1}{1 - \cot x} dx\]
\[\int x \sin x \cos x\ dx\]
\[\int\sqrt{3 - x^2} \text{ dx}\]
\[\int\sqrt{x^2 - 2x} \text{ dx}\]
\[\int\left( x + 1 \right) \sqrt{x^2 - x + 1} \text{ dx}\]
\[\int\left( 4x + 1 \right) \sqrt{x^2 - x - 2} \text{ dx }\]
\[\int\frac{x}{\left( x^2 - a^2 \right) \left( x^2 - b^2 \right)} dx\]
\[\int\frac{1}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)} dx\]
\[\int\frac{x^2 + 1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{1}{7 + 5 \cos x} dx =\]
\[\int\frac{1}{1 - \cos x - \sin x} dx =\]
\[\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx} \]
\[\int\sqrt{\frac{1 + x}{x}} \text{ dx }\]
\[\int\frac{1}{\sin x \left( 2 + 3 \cos x \right)} \text{ dx }\]
\[\int\frac{\sin^2 x}{\cos^6 x} \text{ dx }\]
