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

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

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

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

\frac{x ^{2}}{2} + \frac{3 x ^{4}}{4} + \frac{x ^{6}}{2} + \frac{x ^{8}}{8} + C

Solution steps

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

Expand

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

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

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

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

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

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

= \frac{x ^{2}}{2} + \frac{3 x ^{4}}{4} + \frac{x ^{6}}{2} + \frac{x ^{8}}{8}

Add a constant to the solution

= \frac{x ^{2}}{2} + \frac{3 x ^{4}}{4} + \frac{x ^{6}}{2} + \frac{x ^{8}}{8} + C

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