What is the integral of x^nInx ?

The integral of x^nInx is (Inx^{n+2})/(n+2)+C

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integral of x^nInx

\int x^{n} I n x d x

\frac{I n x ^{n + 2}}{n + 2} + C

Solution steps

\int x^{n} I n x d x

Take the constant out:

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

= I n \cdot \int x x^{n} d x

Simplify

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

= I n \cdot \int x^{1 + n} d x

Apply the Power Rule

: \frac{x ^{n + 2}}{n + 2}

= I n \frac{x ^{n + 2}}{n + 2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{I n x ^{n + 2}}{n + 2}

Add a constant to the solution

= \frac{I n x ^{n + 2}}{n + 2} + C

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