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受欢迎的微积分 >

integral of ysqrt(9-y)

解答

\int y 9 - y d y

解答

\frac{2}{5} (9 - y)^{\frac{5}{2}} - 6 (9 - y)^{\frac{3}{2}} + C

求解步骤

\int y 9 - y d y

使用换元积分法

= \int - (- u + 9) u d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int (- u + 9) u d u

乘开

(- u + 9) u : - u^{\frac{3}{2}} + 9 u

= - \int - u^{\frac{3}{2}} + 9 u d u

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 u^{\frac{3}{2}} d u + \int 9 u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 9 u d u = 6 u^{\frac{3}{2}}

= - (- \frac{2}{5} u^{\frac{5}{2}} + 6 u^{\frac{3}{2}})

u = 9 - y

代回

= - (- \frac{2}{5} (9 - y)^{\frac{5}{2}} + 6 (9 - y)^{\frac{3}{2}})

化简

= \frac{2}{5} (9 - y)^{\frac{5}{2}} - 6 (9 - y)^{\frac{3}{2}}

解答补常数

= \frac{2}{5} (9 - y)^{\frac{5}{2}} - 6 (9 - y)^{\frac{3}{2}} + C

作图

流行的例子

limit as x approaches 3-of 5/(x^2-9)

x \to 3 - lim (\frac{5}{x ^{2} - 9})

limit as x approaches 3 of e^{7/(x-3)}

x \to 3 lim (e^{\frac{7}{x - 3}})

limit as x approaches 1 of 7/((x-1)^2)

x \to 1 lim (\frac{7}{( x - 1 ) ^{2}})

integral of 5/((x^2))

\int \frac{5}{( x ^{2} )} d x

integral of 1/(y-y^2)

\int \frac{1}{y - y ^{2}} d y