What is the integral of (nx)^{(n-1)/n} ?

The integral of (nx)^{(n-1)/n} is (n^{(2n-1)/n}x^{(2n-1)/n})/(2n-1)+C

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integral of (nx)^{(n-1)/n}

\int (n x)^{\frac{n - 1}{n}} d x

\frac{n ^{\frac{2 n - 1}{n}} x ^{\frac{2 n - 1}{n}}}{2 n - 1} + C

Solution steps

\int (n x)^{\frac{n - 1}{n}} d x

(n x)^{\frac{n - 1}{n}} = n^{\frac{n - 1}{n}} x^{\frac{n - 1}{n}}

= \int n^{\frac{n - 1}{n}} x^{\frac{n - 1}{n}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= n^{\frac{n - 1}{n}} \cdot \int x^{\frac{n - 1}{n}} d x

Apply the Power Rule

: \frac{n x ^{\frac{2 n - 1}{n}}}{2 n - 1}

= n^{\frac{n - 1}{n}} \frac{n x ^{\frac{2 n - 1}{n}}}{2 n - 1}

Simplify

n^{\frac{n - 1}{n}} \frac{n x ^{\frac{2 n - 1}{n}}}{2 n - 1} : \frac{n ^{\frac{2 n - 1}{n}} x ^{\frac{2 n - 1}{n}}}{2 n - 1}

= \frac{n ^{\frac{2 n - 1}{n}} x ^{\frac{2 n - 1}{n}}}{2 n - 1}

Add a constant to the solution

= \frac{n ^{\frac{2 n - 1}{n}} x ^{\frac{2 n - 1}{n}}}{2 n - 1} + C

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What is the integral of (nx)^{(n-1)/n} ?
The integral of (nx)^{(n-1)/n} is (n^{(2n-1)/n}x^{(2n-1)/n})/(2n-1)+C