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

The integral of x^3(x^2+3x+1) is (x^6)/6+(3x^5)/5+(x^4)/4+C

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

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

\frac{x ^{6}}{6} + \frac{3 x ^{5}}{5} + \frac{x ^{4}}{4} + C

Solution steps

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

Expand

x^{3} (x^{2} + 3 x + 1) : x^{5} + 3 x^{4} + x^{3}

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

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

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

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

= \frac{x ^{6}}{6} + \frac{3 x ^{5}}{5} + \frac{x ^{4}}{4}

Add a constant to the solution

= \frac{x ^{6}}{6} + \frac{3 x ^{5}}{5} + \frac{x ^{4}}{4} + C

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What is the integral of x^3(x^2+3x+1) ?
The integral of x^3(x^2+3x+1) is (x^6)/6+(3x^5)/5+(x^4)/4+C