Question

find the indefinite integral and check the result by differentiation. (remember the constant of integration.)

$$ int sqrt 5 { x ^ { 3 } } d x $$

Explanation:

Step1: Rewrite the integrand

Rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$. So the integral becomes $\int x^{\frac{3}{5}}dx$.

Step2: Apply 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 $n=\frac{3}{5}$, then $\int x^{\frac{3}{5}}dx=\frac{x^{\frac{3}{5}+1}}{\frac{3}{5}+1}+C$.
Calculate $\frac{3}{5}+1=\frac{3 + 5}{5}=\frac{8}{5}$. So the integral is $\frac{x^{\frac{8}{5}}}{\frac{8}{5}}+C=\frac{5}{8}x^{\frac{8}{5}}+C$.

Step3: Check by differentiation

Differentiate $y = \frac{5}{8}x^{\frac{8}{5}}+C$ using the power rule for differentiation $\frac{d}{dx}(x^{n})=nx^{n - 1}$.
$\frac{dy}{dx}=\frac{5}{8}\times\frac{8}{5}x^{\frac{8}{5}-1}=x^{\frac{3}{5}}=\sqrt[5]{x^{3}}$.

Answer:

$\frac{5}{8}x^{\frac{8}{5}}+C$