What is the integral of (x^2+5)^3 ?

The integral of (x^2+5)^3 is (x^7)/7+3x^5+25x^3+125x+C

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integral of (x^2+5)^3

\int (x^{2} + 5)^{3} d x

\frac{x ^{7}}{7} + 3 x^{5} + 25 x^{3} + 125 x + C

Solution steps

\int (x^{2} + 5)^{3} d x

Expand

(x^{2} + 5)^{3} : x^{6} + 15 x^{4} + 75 x^{2} + 125

= \int x^{6} + 15 x^{4} + 75 x^{2} + 125 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 15 x^{4} d x + \int 75 x^{2} d x + \int 125 d x

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

\int 15 x^{4} d x = 3 x^{5}

\int 75 x^{2} d x = 25 x^{3}

\int 125 d x = 125 x

= \frac{x ^{7}}{7} + 3 x^{5} + 25 x^{3} + 125 x

Add a constant to the solution

= \frac{x ^{7}}{7} + 3 x^{5} + 25 x^{3} + 125 x + C

What is the integral of (x^2+5)^3 ?
The integral of (x^2+5)^3 is (x^7)/7+3x^5+25x^3+125x+C