Question

evaluate the following indefinite integral. ∫8x dx ∫8x dx=□

Explanation:

Step1: Apply the constant multiple rule

The constant multiple rule states that \(\int kf(x)dx = k\int f(x)dx\) where \(k = 8\) and \(f(x)=x\). So, \(\int 8x dx=8\int xdx\).

Step2: Use the power rule for integration

The power rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). Here \(n = 1\), so \(\int xdx=\frac{x^{1+1}}{1 + 1}+C=\frac{x^{2}}{2}+C\).

Step3: Multiply by the constant

Substitute \(\int xdx=\frac{x^{2}}{2}+C\) into \(8\int xdx\). We get \(8\times\frac{x^{2}}{2}+C\).
Simplify \(8\times\frac{x^{2}}{2}\) to \(4x^{2}\).

Answer:

\(4x^{2}+C\)