What is the integral of (3x^2+5x+1/(2x)+1) ?

The integral of (3x^2+5x+1/(2x)+1) is x^3+(5x^2)/2+1/2 ln|x|+x+C

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integral of (3x^2+5x+1/(2x)+1)

\int (3 x^{2} + 5 x + \frac{1}{2 x} + 1) d x

x^{3} + \frac{5 x ^{2}}{2} + \frac{1}{2} ln ∣ x ∣ + x + C

Solution steps

\int 3 x^{2} + 5 x + \frac{1}{2 x} + 1 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 3 x^{2} d x + \int 5 x d x + \int \frac{1}{2 x} d x + \int 1 d x

\int 3 x^{2} d x = x^{3}

\int 5 x d x = \frac{5 x ^{2}}{2}

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int 1 d x = x

= x^{3} + \frac{5 x ^{2}}{2} + \frac{1}{2} ln ∣ x ∣ + x

Add a constant to the solution

= x^{3} + \frac{5 x ^{2}}{2} + \frac{1}{2} ln ∣ x ∣ + x + C

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What is the integral of (3x^2+5x+1/(2x)+1) ?
The integral of (3x^2+5x+1/(2x)+1) is x^3+(5x^2)/2+1/2 ln|x|+x+C