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Popular >

integral of x^7ln(1+x)

Solution

\int x^{7} ln (1 + x) d x

Solution

\frac{1}{8} x^{8} ln (1 + x) - \frac{1}{8} (\frac{x ^{8}}{8} - \frac{x ^{7}}{7} + \frac{x ^{6}}{6} - \frac{x ^{5}}{5} + \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣) + C

Solution steps

\int x^{7} ln (1 + x) d x

Apply Integration By Parts

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

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

\int \frac{x ^{8}}{8 ( 1 + x )} d x = \frac{1}{8} (\frac{x ^{8}}{8} - \frac{x ^{7}}{7} + \frac{x ^{6}}{6} - \frac{x ^{5}}{5} + \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣)

= \frac{1}{8} x^{8} ln (1 + x) - \frac{1}{8} (\frac{x ^{8}}{8} - \frac{x ^{7}}{7} + \frac{x ^{6}}{6} - \frac{x ^{5}}{5} + \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣)

Add a constant to the solution

= \frac{1}{8} x^{8} ln (1 + x) - \frac{1}{8} (\frac{x ^{8}}{8} - \frac{x ^{7}}{7} + \frac{x ^{6}}{6} - \frac{x ^{5}}{5} + \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣) + C

Graph

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Frequently Asked Questions (FAQ)

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