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Question
\[\int\frac{x}{\sqrt{x^4 + a^4}} dx\]
Sum
Solution
` ∫ {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}`
\[ = \frac{1}{2}\int\frac{dt}{\sqrt{t^2 + \left( a^2 \right)^2}}\]
\[ = \frac{1}{2} \text{ log }\left| t + \sqrt{t^2 + a^4} \right| + C\]
\[ = \frac{1}{2} \text{ log }\left| x^2 + \sqrt{x^4 + a^4} \right| + C\]
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