What is the integral of (5x+3)(x^2-9) ?

The integral of (5x+3)(x^2-9) is (5x^4)/4-(45x^2)/2+x^3-27x+C

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integral of (5x+3)(x^2-9)

\int (5 x + 3) (x^{2} - 9) d x

\frac{5 x ^{4}}{4} - \frac{45 x ^{2}}{2} + x^{3} - 27 x + C

Solution steps

\int (5 x + 3) (x^{2} - 9) d x

Expand

(5 x + 3) (x^{2} - 9) : 5 x^{3} - 45 x + 3 x^{2} - 27

= \int 5 x^{3} - 45 x + 3 x^{2} - 27 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 x^{3} d x - \int 45 x d x + \int 3 x^{2} d x - \int 27 d x

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

\int 45 x d x = \frac{45 x ^{2}}{2}

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

\int 27 d x = 27 x

= \frac{5 x ^{4}}{4} - \frac{45 x ^{2}}{2} + x^{3} - 27 x

Add a constant to the solution

= \frac{5 x ^{4}}{4} - \frac{45 x ^{2}}{2} + x^{3} - 27 x + C

\frac{d}{d x} (x^{2} + y^{2} + x y)

y = 2 x - 2 x^{2}, [0, 1]

y = \frac{1}{x ^{2}}, y = 0, x = 1, x = 5

x \to 0 lim (\frac{sin ( 2 x )}{x} + 4)

\frac{d}{d x} ((2 x^{2} - 4 x)^{\frac{1}{2}})

What is the integral of (5x+3)(x^2-9) ?
The integral of (5x+3)(x^2-9) is (5x^4)/4-(45x^2)/2+x^3-27x+C