What is the integral of 36(x^2+1)(x^3+3x)^2 ?

The integral of 36(x^2+1)(x^3+3x)^2 is 36((x^9)/9+x^7+3x^5+3x^3)+C

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integral of 36(x^2+1)(x^3+3x)^2

\int 36 (x^{2} + 1) (x^{3} + 3 x)^{2} d x

36 (\frac{x ^{9}}{9} + x^{7} + 3 x^{5} + 3 x^{3}) + C

Solution steps

\int 36 (x^{2} + 1) (x^{3} + 3 x)^{2} d x

Take the constant out:

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

= 36 \cdot \int (x^{2} + 1) (x^{3} + 3 x)^{2} d x

Expand

(x^{2} + 1) (x^{3} + 3 x)^{2} : x^{8} + 7 x^{6} + 15 x^{4} + 9 x^{2}

= 36 \cdot \int x^{8} + 7 x^{6} + 15 x^{4} + 9 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

= 36 (\int x^{8} d x + \int 7 x^{6} d x + \int 15 x^{4} d x + \int 9 x^{2} d x)

\int x^{8} d x = \frac{x ^{9}}{9}

\int 7 x^{6} d x = x^{7}

\int 15 x^{4} d x = 3 x^{5}

\int 9 x^{2} d x = 3 x^{3}

= 36 (\frac{x ^{9}}{9} + x^{7} + 3 x^{5} + 3 x^{3})

Add a constant to the solution

= 36 (\frac{x ^{9}}{9} + x^{7} + 3 x^{5} + 3 x^{3}) + C

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What is the integral of 36(x^2+1)(x^3+3x)^2 ?
The integral of 36(x^2+1)(x^3+3x)^2 is 36((x^9)/9+x^7+3x^5+3x^3)+C