integral of (2x+5y)^6

解答

\int (2 x + 5 y)^{6} d y

64 x^{6} y + 480 x^{5} y^{2} + 2000 x^{4} y^{3} + 5000 x^{3} y^{4} + 7500 x^{2} y^{5} + 6250 x y^{6} + \frac{15625 y ^{7}}{7} + C

求解步骤

\int (2 x + 5 y)^{6} d y

乘开

(2 x + 5 y)^{6} : 64 x^{6} + 960 x^{5} y + 6000 x^{4} y^{2} + 20000 x^{3} y^{3} + 37500 x^{2} y^{4} + 37500 x y^{5} + 15625 y^{6}

= \int 64 x^{6} + 960 x^{5} y + 6000 x^{4} y^{2} + 20000 x^{3} y^{3} + 37500 x^{2} y^{4} + 37500 x y^{5} + 15625 y^{6} d y

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 64 x^{6} d y + \int 960 x^{5} y d y + \int 6000 x^{4} y^{2} d y + \int 20000 x^{3} y^{3} d y + \int 37500 x^{2} y^{4} d y + \int 37500 x y^{5} d y + \int 15625 y^{6} d y

\int 64 x^{6} d y = 64 x^{6} y

\int 960 x^{5} y d y = 480 x^{5} y^{2}

\int 6000 x^{4} y^{2} d y = 2000 x^{4} y^{3}

\int 20000 x^{3} y^{3} d y = 5000 x^{3} y^{4}

\int 37500 x^{2} y^{4} d y = 7500 x^{2} y^{5}

\int 37500 x y^{5} d y = 6250 x y^{6}

\int 15625 y^{6} d y = \frac{15625 y ^{7}}{7}

= 64 x^{6} y + 480 x^{5} y^{2} + 2000 x^{4} y^{3} + 5000 x^{3} y^{4} + 7500 x^{2} y^{5} + 6250 x y^{6} + \frac{15625 y ^{7}}{7}

解答补常数

= 64 x^{6} y + 480 x^{5} y^{2} + 2000 x^{4} y^{3} + 5000 x^{3} y^{4} + 7500 x^{2} y^{5} + 6250 x y^{6} + \frac{15625 y ^{7}}{7} + C

\int_{0}^{1} 2 π (x + 5) x^{2} d x

x \to 4 lim (8 - 3 x)

e^{2} x \frac{d y}{d x} - e^{x} y = e^{(- x)} y^{2}

d er i v a t i v e f (x) = 2 e^{2}

\frac{d x}{d t} = sin (π x)