integral of (5x-3)^2(2x-1)

解答

\int (5 x - 3)^{2} (2 x - 1) d x

\frac{50 x ^{4}}{3} - \frac{25 x ^{4}}{6} - 20 x^{3} - \frac{25 x ^{3}}{3} + 24 x^{2} - 9 x + C

求解步骤

\int (5 x - 3)^{2} (2 x - 1) d x

使用分布积分法

= (2 x - 1) (\frac{25 x ^{3}}{3} - 15 x^{2} + 9 x) - \int 2 (\frac{25 x ^{3}}{3} - 15 x^{2} + 9 x) d x

\int 2 (\frac{25 x ^{3}}{3} - 15 x^{2} + 9 x) d x = \frac{25 x ^{4}}{6} - 10 x^{3} + 9 x^{2}

= (2 x - 1) (\frac{25 x ^{3}}{3} - 15 x^{2} + 9 x) - (\frac{25 x ^{4}}{6} - 10 x^{3} + 9 x^{2})

化简

(2 x - 1) (\frac{25 x ^{3}}{3} - 15 x^{2} + 9 x) - (\frac{25 x ^{4}}{6} - 10 x^{3} + 9 x^{2}) : \frac{50 x ^{4}}{3} - \frac{25 x ^{4}}{6} - 20 x^{3} - \frac{25 x ^{3}}{3} + 24 x^{2} - 9 x

= \frac{50 x ^{4}}{3} - \frac{25 x ^{4}}{6} - 20 x^{3} - \frac{25 x ^{3}}{3} + 24 x^{2} - 9 x

解答补常数

= \frac{50 x ^{4}}{3} - \frac{25 x ^{4}}{6} - 20 x^{3} - \frac{25 x ^{3}}{3} + 24 x^{2} - 9 x + C

x \to 0 lim (x ln (1 + \frac{1}{x}))

d er i v a t i v e \frac{- 3 x + 12}{2 6 - x}

d er i v a t i v e f (x) = (tan (\frac{1}{x}))^{s e c (\frac{1}{x})}

\int \frac{x}{( 1 + x ^{2} ) ^{y}} d x

\frac{d}{d x} ((x - 1) (x + 3))