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

The integral of 3x(5-x) is (15x^2)/2-x^3+C

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integral of 3x(5-x)

\int 3 x (5 - x) d x

\frac{15 x ^{2}}{2} - x^{3} + C

Solution steps

\int 3 x (5 - x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x (5 - x) d x

Expand

x (5 - x) : 5 x - x^{2}

= 3 \cdot \int 5 x - 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

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

\int 5 x d x = \frac{5 x ^{2}}{2}

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

= 3 (\frac{5 x ^{2}}{2} - \frac{x ^{3}}{3})

Simplify

3 (\frac{5 x ^{2}}{2} - \frac{x ^{3}}{3}) : \frac{15 x ^{2}}{2} - x^{3}

= \frac{15 x ^{2}}{2} - x^{3}

Add a constant to the solution

= \frac{15 x ^{2}}{2} - x^{3} + C

x \to 0 lim (\frac{x}{2 tan ( x )})

\frac{d}{d x} (\frac{x ^{3}}{6})

\int_{0}^{1} \frac{1}{5 x - 3} d x

2 y^{^{''}} + y^{^{'}} - y = x + 1

x \to 3 lim (x^{2} + 4 x - 3)