What is the integral of x^7ln(4x^4) ?

The integral of x^7ln(4x^4) is 1/4 x^8(ln(2)+2ln(x))-(x^8)/(16)+C

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integral of x^7ln(4x^4)

\int x^{7} ln (4 x^{4}) d x

\frac{1}{4} x^{8} (ln (2) + 2 ln (x)) - \frac{x ^{8}}{16} + C

Solution steps

\int x^{7} ln (4 x^{4}) d x

Apply Integration By Parts

: \frac{1}{8} x^{8} ln (4 x^{4}) - \int \frac{x ^{7}}{2} d x

= \frac{1}{8} x^{8} ln (4 x^{4}) - \int \frac{x ^{7}}{2} d x

\int \frac{x ^{7}}{2} d x = \frac{x ^{8}}{16}

= \frac{1}{8} x^{8} ln (4 x^{4}) - \frac{x ^{8}}{16}

\frac{1}{8} x^{8} ln (4 x^{4}) = \frac{1}{4} x^{8} (ln (2) + 2 ln (x))

= \frac{1}{4} x^{8} (ln (2) + 2 ln (x)) - \frac{x ^{8}}{16}

Add a constant to the solution

= \frac{1}{4} x^{8} (ln (2) + 2 ln (x)) - \frac{x ^{8}}{16} + C

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What is the integral of x^7ln(4x^4) ?
The integral of x^7ln(4x^4) is 1/4 x^8(ln(2)+2ln(x))-(x^8)/(16)+C