What is the integral of ((-1)^{n+1}(x-7)^n)/n ?

The integral of ((-1)^{n+1}(x-7)^n)/n is ((-1)^{n+1})/n*((x-7)^{n+1})/(n+1)+C

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integral of ((-1)^{n+1}(x-7)^n)/n

\int \frac{( - 1 ) ^{n + 1} ( x - 7 ) ^{n}}{n} d x

\frac{( - 1 ) ^{n + 1}}{n} \cdot \frac{( x - 7 ) ^{n + 1}}{n + 1} + C

Solution steps

\int \frac{( - 1 ) ^{n + 1} ( x - 7 ) ^{n}}{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 \int (x - 7)^{n} d x

Apply u-substitution

: \int u^{n} d u

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

Apply the Power Rule

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

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

Substitute back

u = x - 7

= \frac{( - 1 ) ^{n + 1}}{n} \cdot \frac{( x - 7 ) ^{n + 1}}{n + 1}

Add a constant to the solution

= \frac{( - 1 ) ^{n + 1}}{n} \cdot \frac{( x - 7 ) ^{n + 1}}{n + 1} + C

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What is the integral of ((-1)^{n+1}(x-7)^n)/n ?
The integral of ((-1)^{n+1}(x-7)^n)/n is ((-1)^{n+1})/n*((x-7)^{n+1})/(n+1)+C