Question

find the indefinite integral and check your result by differentiation. (remember the constant of integration.)
int ( x + 8 ) d x

Explanation:

Step1: Apply sum rule of integration

The sum rule states that \(\int(f(x)+g(x))dx=\int f(x)dx+\int g(x)dx\). So, \(\int(x + 8)dx=\int xdx+\int8dx\).

Step2: Integrate each term

• For \(\int xdx\), using the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n

eq - 1\)), when \(n = 1\), \(\int xdx=\frac{x^{1+1}}{1 + 1}=\frac{x^{2}}{2}\).

• For \(\int8dx\), since \(\int kdx=kx + C\) (\(k\) is a constant), \(\int8dx=8x\).

Combining these results, \(\int(x + 8)dx=\frac{x^{2}}{2}+8x+C\).

Step3: Check by differentiation

Differentiate \(y=\frac{x^{2}}{2}+8x+C\) using the sum rule of differentiation \((u + v+w)'=u'+v'+w'\).

• The derivative of \(\frac{x^{2}}{2}\) using the power rule \((x^n)'=nx^{n - 1}\) is \(\frac{2x^{2-1}}{2}=x\).
• The derivative of \(8x\) is \(8\).
• The derivative of the constant \(C\) is \(0\).

So, \(y'=x + 8\), which is the integrand.

Answer:

\(\frac{x^{2}}{2}+8x+C\)