Question

evaluate the integral
int_{5}^{7} 9 , dx

int_{5}^{7} 9 , dx = (simplify your answer.)

Explanation:

Step1: Apply the integral rule for a constant

The integral of a constant \(k\) with respect to \(x\) is \(k\times x + C\). For \(\int 9dx\), the antiderivative is \(9x\).

Step2: Use the fundamental theorem of calculus

The fundamental theorem of calculus states that \(\int_{a}^{b}f(x)dx=F(b)-F(a)\), where \(F(x)\) is the antiderivative of \(f(x)\). Here, \(a = 5\), \(b = 7\), and \(F(x)=9x\). So we calculate \(9\times7-9\times5\).

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Answer:

\(18\)