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

The integral of 3x^2(x^3-5)^2 is (x^9)/3-5x^6+25x^3+C

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

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

\frac{x ^{9}}{3} - 5 x^{6} + 25 x^{3} + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x^{2} (x^{3} - 5)^{2} d x

Expand

x^{2} (x^{3} - 5)^{2} : x^{8} - 10 x^{5} + 25 x^{2}

= 3 \cdot \int x^{8} - 10 x^{5} + 25 x^{2} 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

= 3 (\int x^{8} d x - \int 10 x^{5} d x + \int 25 x^{2} d x)

\int x^{8} d x = \frac{x ^{9}}{9}

\int 10 x^{5} d x = \frac{5 x ^{6}}{3}

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

= 3 (\frac{x ^{9}}{9} - \frac{5 x ^{6}}{3} + \frac{25 x ^{3}}{3})

Simplify

3 (\frac{x ^{9}}{9} - \frac{5 x ^{6}}{3} + \frac{25 x ^{3}}{3}) : \frac{x ^{9}}{3} - 5 x^{6} + 25 x^{3}

= \frac{x ^{9}}{3} - 5 x^{6} + 25 x^{3}

Add a constant to the solution

= \frac{x ^{9}}{3} - 5 x^{6} + 25 x^{3} + C

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\int (sec (x))^{3} tan (x) d x

What is the integral of 3x^2(x^3-5)^2 ?
The integral of 3x^2(x^3-5)^2 is (x^9)/3-5x^6+25x^3+C