5a. the expression $\frac{3sqrt{x}-5}{sqrt{x}}$ can be written as $3 - 5x^{p}$. write down the value of $p$…

5a. the expression $\frac{3sqrt{x}-5}{sqrt{x}}$ can be written as $3 - 5x^{p}$. write down the value of $p$. 1 mark\n5b. hence, find the value of $int_{1}^{9}(\frac{3sqrt{x}-5}{sqrt{x}})dx$. 4 marks\nconsider the function $f$ defined by $f(x)=6 + 6cos x$, for $0leq xleq4pi$. the following diagram shows the graph of $y = f(x)$.\nthe graph of $f$ touches the $x$-axis at points a and b, as shown. the shaded region is enclosed by the graph of $y = f(x)$ and the $x$-axis, between the points a and b.

5a. the expression $\frac{3sqrt{x}-5}{sqrt{x}}$ can be written as $3 - 5x^{p}$. write down the value of $p$. 1 mark\n5b. hence, find the value of $int_{1}^{9}(\frac{3sqrt{x}-5}{sqrt{x}})dx$. 4 marks\nconsider the function $f$ defined by $f(x)=6 + 6cos x$, for $0leq xleq4pi$. the following diagram shows the graph of $y = f(x)$.\nthe graph of $f$ touches the $x$-axis at points a and b, as shown. the shaded region is enclosed by the graph of $y = f(x)$ and the $x$-axis, between the points a and b.

Answer

Explanation:

Step1: Simplify the expression

We have $\frac{3\sqrt{x}-5}{\sqrt{x}}=\frac{3\sqrt{x}}{\sqrt{x}}-\frac{5}{\sqrt{x}} = 3 - 5x^{-\frac{1}{2}}$. Comparing with $3 - 5x^{p}$, we get $p=-\frac{1}{2}$.

Step2: Calculate the integral

We know that $\int_{1}^{9}\left(\frac{3\sqrt{x}-5}{\sqrt{x}}\right)dx=\int_{1}^{9}(3 - 5x^{-\frac{1}{2}})dx$. Using the integral rules $\int kdx=kx + C$ and $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have: $\int_{1}^{9}(3 - 5x^{-\frac{1}{2}})dx=\left[3x-5\times\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right]{1}^{9}=\left[3x - 10\sqrt{x}\right]{1}^{9}$.

Step3: Evaluate the definite - integral

Substitute the upper and lower limits: $(3\times9-10\sqrt{9})-(3\times1 - 10\sqrt{1})=(27-30)-(3 - 10)$. $=- 3-(-7)=4$.

Answer:

5a. $p =-\frac{1}{2}$ 5b. $4$