Question

2 7. evaluate $int_{1}^{4}x dx$.

Explanation:

Step1: Apply power - rule for integration

The antiderivative of $x$ is $\frac{x^{2}}{2}+C$. For definite integral $\int_{a}^{b}f(x)dx = F(b)-F(a)$, where $F(x)$ is the antiderivative of $f(x)$. Here $f(x)=x$, $a = 1$, $b = 4$ and $F(x)=\frac{x^{2}}{2}$.

Step2: Evaluate $F(b)-F(a)$

Substitute $b = 4$ and $a = 1$ into $F(x)$. We get $F(4)-F(1)=\frac{4^{2}}{2}-\frac{1^{2}}{2}=\frac{16}{2}-\frac{1}{2}=\frac{16 - 1}{2}=\frac{15}{2}$.

Answer:

$\frac{15}{2}$