determine the following indefinite integral. check your work by differentiation.\n int 4sqrt11{x} dx \n int…

Explanation:

Step1: Rewrite the integrand

We know that $\sqrt[11]{x}=x^{\frac{1}{11}}$. So the integral becomes $\int 4x^{\frac{1}{11}}dx$.

Step2: Use the power - rule for integration

The power - rule for integration is $\int ax^{n}dx=\frac{a}{n + 1}x^{n+1}+C$ ($n\neq - 1$). Here $a = 4$ and $n=\frac{1}{11}$. Then $\int 4x^{\frac{1}{11}}dx=4\times\frac{1}{\frac{1}{11}+1}x^{\frac{1}{11}+1}+C$.

Step3: Simplify the coefficient

First, calculate $\frac{1}{\frac{1}{11}+1}=\frac{1}{\frac{1 + 11}{11}}=\frac{11}{12}$. So $4\times\frac{1}{\frac{1}{11}+1}x^{\frac{1}{11}+1}+C=\frac{4\times11}{12}x^{\frac{12}{11}}+C=\frac{11}{3}x^{\frac{12}{11}}+C$.

Step4: Check by differentiation

Differentiate $\frac{11}{3}x^{\frac{12}{11}}+C$ using the power - rule for differentiation $\frac{d}{dx}(ax^{n})=anx^{n - 1}$. We have $\frac{d}{dx}(\frac{11}{3}x^{\frac{12}{11}}+C)=\frac{11}{3}\times\frac{12}{11}x^{\frac{12}{11}-1}=4x^{\frac{1}{11}}$.

Answer:

$\frac{11}{3}x^{\frac{12}{11}}+C$