What is the integral of 8x^7ln(x) ?

The integral of 8x^7ln(x) is x^8ln(x)-(x^8)/8+C

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integral of 8x^7ln(x)

\int 8 x^{7} ln (x) d x

x^{8} ln (x) - \frac{x ^{8}}{8} + C

Solution steps

\int 8 x^{7} ln (x) d x

Take the constant out:

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

= 8 \cdot \int x^{7} ln (x) d x

Apply Integration By Parts

: \frac{1}{8} x^{8} ln (x) - \int \frac{x ^{7}}{8} d x

= 8 (\frac{1}{8} x^{8} ln (x) - \int \frac{x ^{7}}{8} d x)

\int \frac{x ^{7}}{8} d x = \frac{x ^{8}}{64}

= 8 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64})

Simplify

8 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64}) : x^{8} ln (x) - \frac{x ^{8}}{8}

= x^{8} ln (x) - \frac{x ^{8}}{8}

Add a constant to the solution

= x^{8} ln (x) - \frac{x ^{8}}{8} + C

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What is the integral of 8x^7ln(x) ?
The integral of 8x^7ln(x) is x^8ln(x)-(x^8)/8+C