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

The integral of (x^2+2)^3 is (x^7)/7+(6x^5)/5+4x^3+8x+C

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integral of (x^2+2)^3

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

\frac{x ^{7}}{7} + \frac{6 x ^{5}}{5} + 4 x^{3} + 8 x + C

Solution steps

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

Expand

(x^{2} + 2)^{3} : x^{6} + 6 x^{4} + 12 x^{2} + 8

= \int x^{6} + 6 x^{4} + 12 x^{2} + 8 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 6 x^{4} d x + \int 12 x^{2} d x + \int 8 d x

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

\int 6 x^{4} d x = \frac{6 x ^{5}}{5}

\int 12 x^{2} d x = 4 x^{3}

\int 8 d x = 8 x

= \frac{x ^{7}}{7} + \frac{6 x ^{5}}{5} + 4 x^{3} + 8 x

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{6 x ^{5}}{5} + 4 x^{3} + 8 x + C

x \to 0 lim ((x - 4)^{3})

y^{^{'}} = \frac{( y ^{2} + 1 )}{x}

x \to 1 lim (\frac{1 - \frac{2 l n ( 1 + x )}{x}}{x})

\int \frac{12}{x} + 12 x d x

y = \frac{1}{x ^{2}}, y = 9, x = 8

What is the integral of (x^2+2)^3 ?
The integral of (x^2+2)^3 is (x^7)/7+(6x^5)/5+4x^3+8x+C