What is the integral of (x^2+3x+1)/x ?

The integral of (x^2+3x+1)/x is (x^2)/2+3x+ln|x|+C

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integral of (x^2+3x+1)/x

\int \frac{x ^{2} + 3 x + 1}{x} d x

\frac{x ^{2}}{2} + 3 x + ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{2} + 3 x + 1}{x} d x

Expand

\frac{x ^{2} + 3 x + 1}{x} : x + 3 + \frac{1}{x}

= \int x + 3 + \frac{1}{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 x d x + \int 3 d x + \int \frac{1}{x} d x

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

\int 3 d x = 3 x

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{x ^{2}}{2} + 3 x + ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 3 x + ln ∣ x ∣ + C

\int \frac{1}{3 x ^{2} + 4} d x

f (x) = - 2 x^{4} + 10 x^{2}, at x = - 1

\frac{d}{d x} (1 - \frac{2}{x ^{2}})

\int x^{100} + x^{99} d x

(x^{2} - y^{2}) d x + 2 x y d y = 0

What is the integral of (x^2+3x+1)/x ?
The integral of (x^2+3x+1)/x is (x^2)/2+3x+ln|x|+C