7 Find \( A \) as a function of \( x \), given that \( \frac{\mathrm{d} A}{\mathrm{~d} x}=3(x-1)^{2} \) and that \( A=10 \) when \( x=3 \).

Given that \( \frac{\mathrm{d} A}{\mathrm{~d} x}=3(x-1)^{2} \) and \( A=10 \) when \( x=3 \), we can find the function \( A \) as a function of \( x \) by integrating the given derivative. The derivative of \( A \) with respect to \( x \) is \( \frac{\mathrm{d} A}{\mathrm{~d} x}=3(x-1)^{2} \). To find the function \( A \), we need to integrate the derivative with respect to \( x \): \[ A = \int 3(x-1)^{2} \, \mathrm{d}x \] Now, we can integrate the expression to find the function \( A \) as a function of \( x \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int 3\left(x-1\right)^{2} dx\) - step1: Use properties of integrals: \(3\times \int \left(x-1\right)^{2} dx\) - step2: Use the substitution \(dx=1 dt\) to transform the integral\(:\) \(3\times \int \left(x-1\right)^{2}\times 1 dt\) - step3: Simplify: \(3\times \int \left(x-1\right)^{2} dt\) - step4: Use the substitution \(t=x-1\) to transform the integral\(:\) \(3\times \int t^{2} dt\) - step5: Evaluate the integral: \(3\times \frac{t^{2+1}}{2+1}\) - step6: Simplify: \(3\times \frac{t^{3}}{3}\) - step7: Reduce the fraction: \(1\times t^{3}\) - step8: Multiply the terms: \(t^{3}\) - step9: Substitute back: \(\left(x-1\right)^{3}\) - step10: Add the constant of integral C: \(\left(x-1\right)^{3} + C, C \in \mathbb{R}\) The integral of \( 3(x-1)^{2} \) with respect to \( x \) is \( (x-1)^{3} + C \), where \( C \) is an arbitrary constant. Now, we are given that \( A=10 \) when \( x=3 \). We can use this information to find the value of the constant \( C \). Substitute \( A=10 \) and \( x=3 \) into the function \( A = (x-1)^{3} + C \): \[ 10 = (3-1)^{3} + C \] \[ 10 = 8 + C \] \[ C = 2 \] Therefore, the function \( A \) as a function of \( x \) is \( A = (x-1)^{3} + 2 \).