What is the integral of 5x(12x^3-10x) ?

The integral of 5x(12x^3-10x) is 12x^5-(50x^3)/3+C

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integral of 5x(12x^3-10x)

\int 5 x (12 x^{3} - 10 x) d x

12 x^{5} - \frac{50 x ^{3}}{3} + C

Solution steps

\int 5 x (12 x^{3} - 10 x) d x

Take the constant out:

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

= 5 \cdot \int x (12 x^{3} - 10 x) d x

Expand

x (12 x^{3} - 10 x) : 12 x^{4} - 10 x^{2}

= 5 \cdot \int 12 x^{4} - 10 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

= 5 (\int 12 x^{4} d x - \int 10 x^{2} d x)

\int 12 x^{4} d x = \frac{12 x ^{5}}{5}

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

= 5 (\frac{12 x ^{5}}{5} - \frac{10 x ^{3}}{3})

Simplify

5 (\frac{12 x ^{5}}{5} - \frac{10 x ^{3}}{3}) : 12 x^{5} - \frac{50 x ^{3}}{3}

= 12 x^{5} - \frac{50 x ^{3}}{3}

Add a constant to the solution

= 12 x^{5} - \frac{50 x ^{3}}{3} + C

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What is the integral of 5x(12x^3-10x) ?
The integral of 5x(12x^3-10x) is 12x^5-(50x^3)/3+C