Advertisements
Advertisements
Question
` ∫ { x^2 dx}/{x^6 - a^6} dx `
Sum
Solution
\[\int\frac{x^2 dx}{x^6 - a^6}\]
\[\text{ let } x^3 = t\]
\[ \Rightarrow 3 x^2 \text{ dx } = dt\]
\[ \Rightarrow x^2 dx = \frac{dt}{3}\]
\[Now, \int\frac{x^2 dx}{x^6 - a^6}\]
\[ = \frac{1}{3}\int\frac{dt}{t^2 - \left( a^3 \right)^2}\]
\[ = \frac{1}{3} \times \frac{1}{2 a^3} \text{ log } \left| \frac{t - a^3}{t + a^3} \right| + C\]
\[ = \frac{1}{6 a^3} \text{ log }\left| \frac{x^3 - a^3}{x^3 + a^3} \right| + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\text{sin mx }\text{cos nx dx m }\neq n\]
\[\int\frac{1 - \sin x}{x + \cos x} dx\]
\[\int\frac{x + 1}{x \left( x + \log x \right)} dx\]
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} dx\]
` ∫ 1 /{x^{1/3} ( x^{1/3} -1)} ` dx
\[\int {cosec}^4 \text{ 3x } \text{ dx } \]
\[\int\frac{1 - 3x}{3 x^2 + 4x + 2}\text{ dx}\]
\[\int\frac{x^2 + x + 1}{x^2 - x} dx\]
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
\[\int\frac{1}{\cos x \left( \sin x + 2 \cos x \right)} dx\]
\[\int\frac{1}{13 + 3 \cos x + 4 \sin x} dx\]
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
\[\int x^3 \text{ log x dx }\]
\[\int\frac{\log \left( \log x \right)}{x} dx\]
\[\int e^x \left( \cot x + \log \sin x \right) dx\]
\[\int\frac{e^x}{x}\left\{ \text{ x }\left( \log x \right)^2 + 2 \log x \right\} dx\]
\[\int\frac{e^x \left( x - 4 \right)}{\left( x - 2 \right)^3} \text{ dx }\]
\[\int\left( 2x - 5 \right) \sqrt{2 + 3x - x^2} \text{ dx }\]
\[\int\frac{5 x^2 - 1}{x \left( x - 1 \right) \left( x + 1 \right)} dx\]
\[\int\frac{x^2 + 6x - 8}{x^3 - 4x} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x + 1 \right)^2} dx\]
\[\int\frac{x}{\left( x + 1 \right) \left( x^2 + 1 \right)} dx\]
\[\int\frac{1}{\left( x + 1 \right)^2 \left( x^2 + 1 \right)} dx\]
Find \[\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)^2}dx\]
\[\int\frac{x^4}{\left( x - 1 \right) \left( x^2 + 1 \right)} dx\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{2x + 3}} \text{ dx }\]
\[\int\frac{1}{\left( 2 x^2 + 3 \right) \sqrt{x^2 - 4}} \text{ dx }\]
\[\int x^{\sin x} \left( \frac{\sin x}{x} + \cos x . \log x \right) dx\] is equal to
\[\int \sec^2 x \cos^2 2x \text{ dx }\]
\[\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx} \]
\[\int\frac{1}{\left( \sin^{- 1} x \right) \sqrt{1 - x^2}} \text{ dx} \]
\[\int \cot^4 x\ dx\]
\[\int\frac{x^2}{\left( x - 1 \right)^3} dx\]
\[\int\frac{\sqrt{a} - \sqrt{x}}{1 - \sqrt{ax}}\text{ dx }\]
\[\int\frac{1}{2 - 3 \cos 2x} \text{ dx }\]
\[\int \sec^6 x\ dx\]
\[\int \sin^{- 1} \sqrt{x}\ dx\]
\[\int\frac{x^2}{\left( x - 1 \right)^3 \left( x + 1 \right)} \text{ dx}\]
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
