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

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

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integral of (1+x^2)^2

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

x + \frac{2 x ^{3}}{3} + \frac{x ^{5}}{5} + C

Solution steps

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

Expand

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

= \int 1 + 2 x^{2} + x^{4} 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 1 d x + \int 2 x^{2} d x + \int x^{4} d x

\int 1 d x = x

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

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

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

Add a constant to the solution

= x + \frac{2 x ^{3}}{3} + \frac{x ^{5}}{5} + C

\int \frac{3 - 6 x ^{3} + 4 x ^{6}}{x ^{6}} d x

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\frac{d}{d x} ((x + 1) (x - 1))

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