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

The integral of (3x^2+5)^3 is (27x^7)/7+27x^5+75x^3+125x+C

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integral of (3x^2+5)^3

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

\frac{27 x ^{7}}{7} + 27 x^{5} + 75 x^{3} + 125 x + C

Solution steps

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

Expand

(3 x^{2} + 5)^{3} : 27 x^{6} + 135 x^{4} + 225 x^{2} + 125

= \int 27 x^{6} + 135 x^{4} + 225 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 27 x^{6} d x + \int 135 x^{4} d x + \int 225 x^{2} d x + \int 125 d x

\int 27 x^{6} d x = \frac{27 x ^{7}}{7}

\int 135 x^{4} d x = 27 x^{5}

\int 225 x^{2} d x = 75 x^{3}

\int 125 d x = 125 x

= \frac{27 x ^{7}}{7} + 27 x^{5} + 75 x^{3} + 125 x

Add a constant to the solution

= \frac{27 x ^{7}}{7} + 27 x^{5} + 75 x^{3} + 125 x + C

What is the integral of (3x^2+5)^3 ?
The integral of (3x^2+5)^3 is (27x^7)/7+27x^5+75x^3+125x+C