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

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

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integral of (x-2)(3x+1)

\int (x - 2) (3 x + 1) d x

\frac{3 x ^{3}}{2} - \frac{x ^{3}}{2} - 3 x^{2} + \frac{x ^{2}}{2} - 2 x + C

Solution steps

\int (x - 2) (3 x + 1) d x

Apply Integration By Parts

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

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

\int 3 (\frac{x ^{2}}{2} - 2 x) d x = 3 (\frac{x ^{3}}{6} - x^{2})

= (3 x + 1) (\frac{x ^{2}}{2} - 2 x) - 3 (\frac{x ^{3}}{6} - x^{2})

Simplify

(3 x + 1) (\frac{x ^{2}}{2} - 2 x) - 3 (\frac{x ^{3}}{6} - x^{2}) : \frac{3 x ^{3}}{2} - \frac{x ^{3}}{2} - 3 x^{2} + \frac{x ^{2}}{2} - 2 x

= \frac{3 x ^{3}}{2} - \frac{x ^{3}}{2} - 3 x^{2} + \frac{x ^{2}}{2} - 2 x

Add a constant to the solution

= \frac{3 x ^{3}}{2} - \frac{x ^{3}}{2} - 3 x^{2} + \frac{x ^{2}}{2} - 2 x + C

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