What is the integral of (1+2x^2)/x ?

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

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integral of (1+2x^2)/x

\int \frac{1 + 2 x ^{2}}{x} d x

ln ∣ x ∣ + x^{2} + C

Solution steps

\int \frac{1 + 2 x ^{2}}{x} d x

Expand

\frac{1 + 2 x ^{2}}{x} : \frac{1}{x} + 2 x

= \int \frac{1}{x} + 2 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

= \int \frac{1}{x} d x + \int 2 x d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + x^{2}

Add a constant to the solution

= ln ∣ x ∣ + x^{2} + C

\int \frac{144}{x ^{3} - 12 x ^{2}} d x

f (x) = 5 cos (x), at x = \frac{π}{3}

cos (x), - cos (x) + 2

f (2) = \frac{x + 11}{x - 9}

\frac{x ^{4}}{2} - \frac{3}{2} x^{2} - x