What is the integral of 30x^2(1-x)^2 ?

The integral of 30x^2(1-x)^2 is 30((x^3)/3-(x^4)/2+(x^5)/5)+C

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integral of 30x^2(1-x)^2

\int 30 x^{2} (1 - x)^{2} d x

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

Solution steps

\int 30 x^{2} (1 - x)^{2} d x

Take the constant out:

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

= 30 \cdot \int x^{2} (1 - x)^{2} d x

Expand

x^{2} (1 - x)^{2} : x^{2} - 2 x^{3} + x^{4}

= 30 \cdot \int x^{2} - 2 x^{3} + x^{4} 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

= 30 (\int x^{2} d x - \int 2 x^{3} d x + \int x^{4} d x)

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

\int 2 x^{3} d x = \frac{x ^{4}}{2}

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

= 30 (\frac{x ^{3}}{3} - \frac{x ^{4}}{2} + \frac{x ^{5}}{5})

Add a constant to the solution

= 30 (\frac{x ^{3}}{3} - \frac{x ^{4}}{2} + \frac{x ^{5}}{5}) + C

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What is the integral of 30x^2(1-x)^2 ?
The integral of 30x^2(1-x)^2 is 30((x^3)/3-(x^4)/2+(x^5)/5)+C