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

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

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

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

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

Solution steps

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

Expand

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

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

\int 5 d x = 5 x

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

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

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

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

Add a constant to the solution

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

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