What is the integral of-1/3 (x/3)^n ?

The integral of-1/3 (x/3)^n is -(x^{n+1})/(3^{1+n)(n+1)}+C

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integral of-1/3 (x/3)^n

\int - \frac{1}{3} (\frac{x}{3})^{n} d x

- \frac{x ^{n + 1}}{3 ^{1 + n} ( n + 1 )} + C

Solution steps

\int - \frac{1}{3} (\frac{x}{3})^{n} d x

Take the constant out:

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

= - \frac{1}{3} \cdot \int (\frac{x}{3})^{n} d x

Apply exponent rule

= - \frac{1}{3} \cdot \int \frac{x ^{n}}{3 ^{n}} d x

Take the constant out:

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

= - \frac{1}{3} \cdot \frac{1}{3 ^{n}} \cdot \int x^{n} d x

Apply the Power Rule

: \frac{x ^{n + 1}}{n + 1}

= - \frac{1}{3} \cdot \frac{1}{3 ^{n}} \cdot \frac{x ^{n + 1}}{n + 1}

Simplify

- \frac{1}{3} \cdot \frac{1}{3 ^{n}} \cdot \frac{x ^{n + 1}}{n + 1} : - \frac{x ^{n + 1}}{3 ^{1 + n} ( n + 1 )}

= - \frac{x ^{n + 1}}{3 ^{1 + n} ( n + 1 )}

Add a constant to the solution

= - \frac{x ^{n + 1}}{3 ^{1 + n} ( n + 1 )} + C

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What is the integral of-1/3 (x/3)^n ?
The integral of-1/3 (x/3)^n is -(x^{n+1})/(3^{1+n)(n+1)}+C