f(x)=4x² - 5x\nf(x)=f(x)\n∫₁⁴ f(x)dx=□\nrelated content
f(x)=4x² - 5x\nf(x)=f(x)\n∫₁⁴ f(x)dx=□\nrelated content
Answer
Explanation:
Step1: Find the derivative of $F(x)$
Using the power - rule $(x^n)'=nx^{n - 1}$, if $F(x)=4x^{2}-5x$, then $f(x)=F'(x)=(4x^{2}-5x)'=8x - 5$.
Step2: Calculate the definite integral
By the fundamental theorem of calculus, $\int_{a}^{b}f(x)dx=F(b)-F(a)$. Here $a = 1$, $b = 4$, and $F(x)=4x^{2}-5x$. $F(4)=4\times4^{2}-5\times4=4\times16 - 20=64 - 20 = 44$. $F(1)=4\times1^{2}-5\times1=4 - 5=-1$. $\int_{1}^{4}f(x)dx=F(4)-F(1)=44-(-1)=45$.
Answer:
$45$