Question

∫ (x+1/x)^2 dx. show step by step solution

Answer 1

Answer on Question #52458 – Math – Integral Calculus

$$\int \left(x + \frac {1}{x}\right) ^ {2} d x$$

Show step by step solution.

Solution

$$\int \left(x + \frac {1}{x}\right) ^ {2} d x = \int \left(x ^ {2} + 2 x * \frac {1}{x} + \frac {1}{x ^ {2}}\right) d x = \int x ^ {2} d x + 2 \int d x + \int \frac {d x}{x ^ {2}} = \frac {x ^ {3}}{3} + 2 x - \frac {1}{x} + C,$$

where $C$ is an arbitrary real constant.

We used the following formulas:

$$\int (f (x) + g (x)) d x = \int f (x) d x + \int g (x) d x,$$ $$\int A f (x) d x = A \int f (x) d x,$$ $$\int x ^ {n} d x = \frac {x ^ {n + 1}}{n + 1} + C, \text{ where } C \text{ is an arbitrary real constant, } n \neq - 1.$$ $$(a + b) ^ {2} = a ^ {2} + 2 a b + b ^ {2}.$$

Answer: $\frac{x^3}{3} + 2x + \frac{1}{x} + C$.

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