Question

evaluate the following indefinite integral.
int 4 x d x
int 4 x d x=square

Explanation:

Step1: Use the constant - multiple rule of integration

The constant - multiple rule states that \(\int kf(x)dx=k\int f(x)dx\), where \(k = 4\) and \(f(x)=x\). So, \(\int4x dx=4\int xdx\).

Step2: Apply the power rule of integration

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

Step3: Substitute back the result of \(\int xdx\)

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

Answer:

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