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

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

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integral of (x^2+1)^2

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

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

Solution steps

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

Expand

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

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

\int \frac{4 x ^{2} + 7 x + 4}{( x ^{2} + 1 ) ^{2}} d x

\frac{d}{d x} (\frac{x ^{3} - 5 x ^{2}}{x ^{8} - 4})

\frac{d}{d x} (1 + sin^{2} (x))

(2 t + 4) i + sin (4 t) j + 2 t^{3} k

x \to 0 + lim (x^{3 x})

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