Question

find the indefinite integral and check your result by differentiation. (remember the constant of integration.)
intleft(x^{5}+5
ight) d x

Explanation:

Step1: Integrate term - by - term

Use the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)) and \(\int a dx=ax + C\) (\(a\) is a constant).
For \(\int x^{5}dx\), with \(n = 5\), we have \(\frac{x^{5+1}}{5 + 1}=\frac{x^{6}}{6}\). For \(\int5dx\), since \(a = 5\), we have \(5x\). So \(\int(x^{5}+5)dx=\int x^{5}dx+\int5dx=\frac{x^{6}}{6}+5x+C\).

Step2: Differentiate the result

Differentiate \(y=\frac{x^{6}}{6}+5x+C\) using the power rule \((x^{n})^\prime=nx^{n - 1}\) and \((C)^\prime = 0\).
\(y^\prime=\frac{6x^{6-1}}{6}+5+0\).
Simplify \(y^\prime=x^{5}+5\).

Answer:

\(\frac{x^{6}}{6}+5x + C\)