What is the integral of 5x(x^2+6)^2 ?

The integral of 5x(x^2+6)^2 is 5((x^6)/6+3x^4+18x^2)+C

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integral of 5x(x^2+6)^2

\int 5 x (x^{2} + 6)^{2} d x

5 (\frac{x ^{6}}{6} + 3 x^{4} + 18 x^{2}) + C

Solution steps

\int 5 x (x^{2} + 6)^{2} 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} + 6)^{2} d x

Expand

x (x^{2} + 6)^{2} : x^{5} + 12 x^{3} + 36 x

= 5 \cdot \int x^{5} + 12 x^{3} + 36 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^{5} d x + \int 12 x^{3} d x + \int 36 x d x)

\int x^{5} d x = \frac{x ^{6}}{6}

\int 12 x^{3} d x = 3 x^{4}

\int 36 x d x = 18 x^{2}

= 5 (\frac{x ^{6}}{6} + 3 x^{4} + 18 x^{2})

Add a constant to the solution

= 5 (\frac{x ^{6}}{6} + 3 x^{4} + 18 x^{2}) + C

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What is the integral of 5x(x^2+6)^2 ?
The integral of 5x(x^2+6)^2 is 5((x^6)/6+3x^4+18x^2)+C