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

The integral of x^3(x^2+1)^2 is (x^8)/8+(x^6)/3+(x^4)/4+C

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

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

\frac{x ^{8}}{8} + \frac{x ^{6}}{3} + \frac{x ^{4}}{4} + C

Solution steps

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

Expand

x^{3} (x^{2} + 1)^{2} : x^{7} + 2 x^{5} + x^{3}

= \int x^{7} + 2 x^{5} + 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^{7} d x + \int 2 x^{5} d x + \int x^{3} d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int 2 x^{5} d x = \frac{x ^{6}}{3}

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

= \frac{x ^{8}}{8} + \frac{x ^{6}}{3} + \frac{x ^{4}}{4}

Add a constant to the solution

= \frac{x ^{8}}{8} + \frac{x ^{6}}{3} + \frac{x ^{4}}{4} + C

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What is the integral of x^3(x^2+1)^2 ?
The integral of x^3(x^2+1)^2 is (x^8)/8+(x^6)/3+(x^4)/4+C