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

The integral of (3x-2)(x+1)^3 is (3x^5)/5+(7x^4)/4+x^3-(3x^2)/2-2x+C

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

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

\frac{3 x ^{5}}{5} + \frac{7 x ^{4}}{4} + x^{3} - \frac{3 x ^{2}}{2} - 2 x + C

Solution steps

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

Expand

(3 x - 2) (x + 1)^{3} : 3 x^{4} + 7 x^{3} + 3 x^{2} - 3 x - 2

= \int 3 x^{4} + 7 x^{3} + 3 x^{2} - 3 x - 2 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^{4} d x + \int 7 x^{3} d x + \int 3 x^{2} d x - \int 3 x d x - \int 2 d x

\int 3 x^{4} d x = \frac{3 x ^{5}}{5}

\int 7 x^{3} d x = \frac{7 x ^{4}}{4}

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

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

\int 2 d x = 2 x

= \frac{3 x ^{5}}{5} + \frac{7 x ^{4}}{4} + x^{3} - \frac{3 x ^{2}}{2} - 2 x

Add a constant to the solution

= \frac{3 x ^{5}}{5} + \frac{7 x ^{4}}{4} + x^{3} - \frac{3 x ^{2}}{2} - 2 x + C

What is the integral of (3x-2)(x+1)^3 ?
The integral of (3x-2)(x+1)^3 is (3x^5)/5+(7x^4)/4+x^3-(3x^2)/2-2x+C