What is the integral of ((100+x^2)/(10x)) ?

The integral of ((100+x^2)/(10x)) is 1/10 (100ln|x|+(x^2)/2)+C

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integral of ((100+x^2)/(10x))

\int (\frac{100 + x ^{2}}{10 x}) d x

\frac{1}{10} (100 ln ∣ x ∣ + \frac{x ^{2}}{2}) + C

Solution steps

\int \frac{100 + x ^{2}}{10 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{10} \cdot \int \frac{100 + x ^{2}}{x} d x

Expand

\frac{100 + x ^{2}}{x} : \frac{100}{x} + x

= \frac{1}{10} \cdot \int \frac{100}{x} + x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{10} (\int \frac{100}{x} d x + \int x d x)

\int \frac{100}{x} d x = 100 ln ∣ x ∣

\int x d x = \frac{x ^{2}}{2}

= \frac{1}{10} (100 ln ∣ x ∣ + \frac{x ^{2}}{2})

Add a constant to the solution

= \frac{1}{10} (100 ln ∣ x ∣ + \frac{x ^{2}}{2}) + C

What is the integral of ((100+x^2)/(10x)) ?
The integral of ((100+x^2)/(10x)) is 1/10 (100ln|x|+(x^2)/2)+C