What is the integral of f(x)=(x^3+x)(x^2+2) ?

The integral of f(x)=(x^3+x)(x^2+2) is (x^6)/6+(3x^4)/4+x^2+C

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integral of f(x)=(x^3+x)(x^2+2)

integral

f (x) = (x^{3} + x) (x^{2} + 2)

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

Solution steps

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

Expand

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

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

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

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

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

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

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

Add a constant to the solution

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

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