integral of (2x+6)ln(x)

解答

\int (2 x + 6) ln (x) d x

x^{2} ln (x) + 6 x ln (x) - \frac{x ^{2}}{2} - 6 x + C

求解步骤

\int (2 x + 6) ln (x) d x

使用分布积分法

= ln (x) (x^{2} + 6 x) - \int x + 6 d x

\int x + 6 d x = \frac{x ^{2}}{2} + 6 x

= ln (x) (x^{2} + 6 x) - (\frac{x ^{2}}{2} + 6 x)

化简

ln (x) (x^{2} + 6 x) - (\frac{x ^{2}}{2} + 6 x) : x^{2} ln (x) + 6 x ln (x) - \frac{x ^{2}}{2} - 6 x

= x^{2} ln (x) + 6 x ln (x) - \frac{x ^{2}}{2} - 6 x

解答补常数

= x^{2} ln (x) + 6 x ln (x) - \frac{x ^{2}}{2} - 6 x + C

\int \frac{x ^{4} - 1}{x + 1} d x

\int \frac{1}{sin ( x ) ( cos ( x ) ) ^{2}} d x

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\int \frac{3 x ^{2} + 3 x}{2 x ^{3} + 3 x ^{2} - 5} d x

\int \frac{5 - 3 y ^{2}}{y} d y