Indefinite Integral

Find the following integral:

    \[\int \left( x + \frac{1}{x} \right)^{2} \, \mathrm{d}x\]

Hint:

    \[\left(x + \frac{1}{x} \right)^{2}=x^{2}+2+\frac{1}{x^{2}}\]

    \[\int \left( x + \frac{1}{x} \right)^{2} \, \mathrm{d}x=\int \left(x^{2}+2+\frac{1}{x^{2}}\right) \, \mathrm{d}x\]

Answer:

    \[\frac{1}{3}x^{3}+2x-\frac{1}{x}+C\]

where C is an arbitrary constant.

Source: R.L. Rosenberg. (1984). Elementary Calculus: Course Notes. Ottawa, Canada: Holt, Rinehart and Winston. (Additional Exercises in Integration: Question (i))

Author: ascklee

Dr. Lee teaches at the Wee Kim Wee School of Communication and Information at the Nanyang Technological University in Singapore. He founded The Mathematics Digital Library in 2013.

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