Question

find the indefinite integral and check the result by differentiation. (remember the constant of integration.)
intleft(x^{3 / 2}+10 x+7
ight) d x

Explanation:

Step1: Integrate term - by - term

Use the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)).
For \(\int x^{\frac{3}{2}}dx\), \(n=\frac{3}{2}\), then \(\int x^{\frac{3}{2}}dx=\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}=\frac{x^{\frac{5}{2}}}{\frac{5}{2}}=\frac{2}{5}x^{\frac{5}{2}}\).
For \(\int10xdx\), \(n = 1\), then \(\int10xdx=10\times\frac{x^{1 + 1}}{1+1}=10\times\frac{x^{2}}{2}=5x^{2}\).
For \(\int7dx\), \(n = 0\), then \(\int7dx=7x\).
So \(\int(x^{\frac{3}{2}}+10x + 7)dx=\frac{2}{5}x^{\frac{5}{2}}+5x^{2}+7x+C\).

Step2: Differentiate the result

Differentiate \(y=\frac{2}{5}x^{\frac{5}{2}}+5x^{2}+7x+C\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(y^\prime=\frac{2}{5}\times\frac{5}{2}x^{\frac{5}{2}-1}+5\times2x^{2 - 1}+7\times1x^{0}+0\).
\(y^\prime=x^{\frac{3}{2}}+10x + 7\).

Answer:

\(\frac{2}{5}x^{\frac{5}{2}}+5x^{2}+7x+C\)