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

The integral of (x^2+1)^3 is (x^7)/7+(3x^5)/5+x^3+x+C

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integral of (x^2+1)^3

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

\frac{x ^{7}}{7} + \frac{3 x ^{5}}{5} + x^{3} + x + C

Solution steps

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

Expand

(x^{2} + 1)^{3} : x^{6} + 3 x^{4} + 3 x^{2} + 1

= \int x^{6} + 3 x^{4} + 3 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^{6} d x + \int 3 x^{4} d x + \int 3 x^{2} d x + \int 1 d x

\int x^{6} d x = \frac{x ^{7}}{7}

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

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

\int 1 d x = x

= \frac{x ^{7}}{7} + \frac{3 x ^{5}}{5} + x^{3} + x

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{3 x ^{5}}{5} + x^{3} + x + C

\int \frac{1}{2 t} d t

f (x) = x^{3}, at x = 2

\int 4 x arctan (x) d x

x \to 9 lim (\frac{1}{x - 9})

\frac{d}{d x} ((x + \frac{2}{x}) (7 - 2 x^{3}))

What is the integral of (x^2+1)^3 ?
The integral of (x^2+1)^3 is (x^7)/7+(3x^5)/5+x^3+x+C