Question

$\int$ (x + 3) (x² + 6x + 10)⁹ dx = ______________________.

Answer 1

$I=\int \left(x+3\right){\left(x^2+6x+10\right)}^{9}dx$

Let $x^2+6x+10=z$

$\Rightarrow \left(2x+6\right)dx=dz$

$\Rightarrow \left(x+3\right)dx=\frac{dz}{2}$

$\therefore I=\int {\left(z\right)}^9\frac{dz}{2}$

$\Rightarrow I=\frac{1}{2}\int z^{9}dz$

$\Rightarrow I=\frac{1}{2}\times \frac{z^{10}}{10}+C$

$\Rightarrow I=\frac{1}{20}{\left(x^2+6x+10\right)}^{10}+C$


$\int$(x + 3)(x² + 6x + 10)⁹ dx = $\underline{\frac{1}{20}{\left(x^2+6x+10\right)}^{10}+C}$.