Question

Evaluate the definite integrals: $\int^2_1 {(4x^3 -5x^2+6x+9)} \, dx$

Answer 1

We know that power of x can be integrate as

$\int {x}^{n} dx = \frac{ {x}^{n + 1} }{n + 1}$
$\int^2_1 {(4x^3 -5x^2+6x+9)} \, dx \\ \\ = \bigg[4 (\frac{ {x}^{4} }{4} ) - 5 (\frac{ {x}^{3} }{3} ) + 6 \frac{ {x}^{2} }{2} + 9x\bigg]^\bm2_{1} \\ \\ =\bigg[ {x}^{4} - \frac{5}{3} {x}^{3} + 3 {x}^{2} + 9x\bigg]^\bm2_{1} \\ \\ = ( {2}^{4} ) - \frac{5}{3} ( {2})^{3} + 3( {2)}^{2} + 9(2) - ( {1}^{4} ) + \frac{5}{3} ( {1})^{3} - 3( {1)}^{2} - 9(1) \\ \\ = 16 - \frac{40}{3} + 12 + 18 - 1 + \frac{5}{3} - 3 - 9 \\ \\ = 33 - \frac{40}{3} + \frac{5}{3} \\ \\ = \frac{99 - 40 + 5}{3} \\ \\ = \frac{64}{3}$
Hope it helps you.