Question

find ( f ). ( f(x)=sqrt{x}(3 + 10x)), ( f(1)=7 )

Explanation:

Step1: Expand the integrand

First, expand $\sqrt{x}(3 + 10x)=3x^{\frac{1}{2}}+10x^{\frac{3}{2}}$.

Step2: Integrate term - by - term

The integral of $3x^{\frac{1}{2}}$ is $3\times\frac{2}{3}x^{\frac{3}{2}} = 2x^{\frac{3}{2}}$ (using the power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C,n
eq - 1$), and the integral of $10x^{\frac{3}{2}}$ is $10\times\frac{2}{5}x^{\frac{5}{2}}=4x^{\frac{5}{2}}$. So, $f'(x)=\int(3x^{\frac{1}{2}}+10x^{\frac{3}{2}})dx=2x^{\frac{3}{2}}+4x^{\frac{5}{2}}+C_1$.

Step3: Find $C_1$

Given $f'(1) = 7$, substitute $x = 1$ into $f'(x)$: $2\times1^{\frac{3}{2}}+4\times1^{\frac{5}{2}}+C_1=7$, which simplifies to $2 + 4+C_1=7$, so $C_1=1$. Then $f'(x)=2x^{\frac{3}{2}}+4x^{\frac{5}{2}}+1$.

Step4: Integrate $f'(x)$ to find $f(x)$

Integrate $f'(x)$ term - by - term. The integral of $2x^{\frac{3}{2}}$ is $2\times\frac{2}{5}x^{\frac{5}{2}}=\frac{4}{5}x^{\frac{5}{2}}$, the integral of $4x^{\frac{5}{2}}$ is $4\times\frac{2}{7}x^{\frac{7}{2}}=\frac{8}{7}x^{\frac{7}{2}}$, and the integral of $1$ is $x$. So $f(x)=\frac{4}{5}x^{\frac{5}{2}}+\frac{8}{7}x^{\frac{7}{2}}+x+C_2$.

Step5: Since no other condition for $f(x)$ is given (assuming the constant of integration $C_2 = 0$ for the most general antiderivative based on the info), we have the final result

$f(x)=\frac{4}{5}x^{\frac{5}{2}}+\frac{8}{7}x^{\frac{7}{2}}+x$

Answer:

$f(x)=\frac{4}{5}x^{\frac{5}{2}}+\frac{8}{7}x^{\frac{7}{2}}+x$