What is the integral of 6x(x^2+1)^2 ?

The integral of 6x(x^2+1)^2 is x^6+3x^4+3x^2+C

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integral of 6x(x^2+1)^2

\int 6 x (x^{2} + 1)^{2} d x

x^{6} + 3 x^{4} + 3 x^{2} + C

Solution steps

\int 6 x (x^{2} + 1)^{2} d x

Take the constant out:

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

= 6 \cdot \int x (x^{2} + 1)^{2} d x

Expand

x (x^{2} + 1)^{2} : x^{5} + 2 x^{3} + x

= 6 \cdot \int x^{5} + 2 x^{3} + 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

= 6 (\int x^{5} d x + \int 2 x^{3} d x + \int x d x)

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

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

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

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

Simplify

6 (\frac{x ^{6}}{6} + \frac{x ^{4}}{2} + \frac{x ^{2}}{2}) : x^{6} + 3 x^{4} + 3 x^{2}

= x^{6} + 3 x^{4} + 3 x^{2}

Add a constant to the solution

= x^{6} + 3 x^{4} + 3 x^{2} + C

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What is the integral of 6x(x^2+1)^2 ?
The integral of 6x(x^2+1)^2 is x^6+3x^4+3x^2+C