Find the integral of int x - 3 x 3 dx - SS3 Mathematics Integral Calculus (Integration) Question
Find the integral of \(\int_{}^{}{(x - 3)(x + 3)}dx\)
\(x^{2} + 9x - 27\)
\(\ {3x}^{2} + 11x - 29\)
\(\ 2x\left\lbrack x^{2} + 9x - 27 \right\rbrack\)
\(\ 3x\lbrack x^{3} + 9x^{2} - 27\rbrack\)
\[\int_{}^{}u\ dv = uv - \ \int_{}^{}v\ du\]
\(u = (x - 3)\), \(\frac{du}{dx} = 1\)
\(\frac{dv}{dx} = (x + 3)\), \(v = \frac{x^{2}}{2} + 3x\)
\(\int_{}^{}{(x - 3)(x + 3)}dx = (x - 3)\left( \frac{x^{2}}{2} + 3x \right) - \int_{}^{}\left( \frac{x^{2}}{2} + 3x \right)(1)\ \)
\[= (x - 3)\left( x^{2} + 6x \right) - \int_{}^{}{x^{2} + 6x}\]
\[= x^{3} + 3x^{2} - 18x - \frac{x^{3}}{3} + \frac{6x^{2}}{2}\]
\[= x^{3} + 3x^{2} - 18x - \frac{x^{3}}{3} + 3x^{2}\]
\[= x^{3} + 6x^{2} - 18x - \frac{x^{3}}{3}\]
\[= 3x^{3} + 18x^{2} - 54x - x^{3}\]
\[= 2x^{3} + 18x^{2} - 54x\]
\(= 2x\lbrack x^{2} + 9x - 27\rbrack\)
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