What is the integral of ((-1)^{n+1}(x-5)^n)/(n5^n) ?

The integral of ((-1)^{n+1}(x-5)^n)/(n5^n) is ((-1)^{n+1})/(n*5^n)*((x-5)^{n+1})/(n+1)+C

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integral of ((-1)^{n+1}(x-5)^n)/(n5^n)

\int \frac{( - 1 ) ^{n + 1} ( x - 5 ) ^{n}}{n 5 ^{n}} d x

\frac{( - 1 ) ^{n + 1}}{n \cdot 5 ^{n}} \cdot \frac{( x - 5 ) ^{n + 1}}{n + 1} + C

Solution steps

\int \frac{( - 1 ) ^{n + 1} ( x - 5 ) ^{n}}{n \cdot 5 ^{n}} d x

Take the constant out:

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

= \frac{( - 1 ) ^{n + 1}}{n \cdot 5 ^{n}} \cdot \int (x - 5)^{n} d x

Apply u-substitution

: \int u^{n} d u

= \frac{( - 1 ) ^{n + 1}}{n \cdot 5 ^{n}} \cdot \int u^{n} d u

Apply the Power Rule

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

= \frac{( - 1 ) ^{n + 1}}{n \cdot 5 ^{n}} \cdot \frac{u ^{n + 1}}{n + 1}

Substitute back

u = x - 5

= \frac{( - 1 ) ^{n + 1}}{n \cdot 5 ^{n}} \cdot \frac{( x - 5 ) ^{n + 1}}{n + 1}

Add a constant to the solution

= \frac{( - 1 ) ^{n + 1}}{n \cdot 5 ^{n}} \cdot \frac{( x - 5 ) ^{n + 1}}{n + 1} + C

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What is the integral of ((-1)^{n+1}(x-5)^n)/(n5^n) ?
The integral of ((-1)^{n+1}(x-5)^n)/(n5^n) is ((-1)^{n+1})/(n*5^n)*((x-5)^{n+1})/(n+1)+C