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

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

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integral of (2x-3)(5x^2-1)

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

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

Solution steps

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

Expand

(2 x - 3) (5 x^{2} - 1) : 10 x^{3} - 2 x - 15 x^{2} + 3

= \int 10 x^{3} - 2 x - 15 x^{2} + 3 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 10 x^{3} d x - \int 2 x d x - \int 15 x^{2} d x + \int 3 d x

\int 10 x^{3} d x = \frac{5 x ^{4}}{2}

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

\int 15 x^{2} d x = 5 x^{3}

\int 3 d x = 3 x

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

Add a constant to the solution

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

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