What is the integral of 5x(x^2-8) ?

The integral of 5x(x^2-8) is 5((x^4)/4-4x^2)+C

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integral of 5x(x^2-8)

\int 5 x (x^{2} - 8) d x

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

Solution steps

\int 5 x (x^{2} - 8) d x

Take the constant out:

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

= 5 \cdot \int x (x^{2} - 8) d x

Expand

x (x^{2} - 8) : x^{3} - 8 x

= 5 \cdot \int x^{3} - 8 x 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

= 5 (\int x^{3} d x - \int 8 x d x)

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

\int 8 x d x = 4 x^{2}

= 5 (\frac{x ^{4}}{4} - 4 x^{2})

Add a constant to the solution

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

\int \frac{1}{c + x ^{2}} d x

\int - 4 x d x

\int (2 x + 8) sin (2 x) d x

\int \frac{x}{cos ^{2} ( 2 x )} d x

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

What is the integral of 5x(x^2-8) ?
The integral of 5x(x^2-8) is 5((x^4)/4-4x^2)+C