evaluate the definite integral. \n∫₀⁶(5x² - 8x + 6)dx\n∫₀⁶(5x² - 8x + 6)dx = \n(simplify your answer.)
evaluate the definite integral. \n∫₀⁶(5x² - 8x + 6)dx\n∫₀⁶(5x² - 8x + 6)dx = \n(simplify your answer.)
Answer
Explanation:
Step1: Find antiderivative
Use power - rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$. $\int(5x^{2}-8x + 6)dx=5\times\frac{x^{3}}{3}-8\times\frac{x^{2}}{2}+6x+C=\frac{5}{3}x^{3}-4x^{2}+6x+C$
Step2: Apply fundamental theorem of calculus
$F(x)=\frac{5}{3}x^{3}-4x^{2}+6x$, then $\int_{0}^{6}(5x^{2}-8x + 6)dx=F(6)-F(0)$. $F(6)=\frac{5}{3}\times6^{3}-4\times6^{2}+6\times6$ $=\frac{5}{3}\times216-4\times36 + 36$ $=360-144 + 36$ $=252$. $F(0)=\frac{5}{3}\times0^{3}-4\times0^{2}+6\times0=0$. So $F(6)-F(0)=252$.
Answer:
$252$