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受欢迎的微积分 >

integral of (1+1/t)^3(1/(t^2))

解答

\int (1 + \frac{1}{t})^{3} (\frac{1}{t ^{2}}) d t

解答

- \frac{( t + 1 ) ^{4}}{4 t ^{4}} + C

求解步骤

\int (1 + \frac{1}{t})^{3} \frac{1}{t ^{2}} d t

使用换元积分法

= \int - u^{3} d u

提出常数:

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

= - \int u^{3} d u

使用幂法则

= - \frac{u ^{4}}{4}

u = 1 + \frac{1}{t}

代回

= - \frac{( 1 + \frac{1}{t} ) ^{4}}{4}

化简

- \frac{( 1 + \frac{1}{t} ) ^{4}}{4} : - \frac{( t + 1 ) ^{4}}{4 t ^{4}}

= - \frac{( t + 1 ) ^{4}}{4 t ^{4}}

解答补常数

= - \frac{( t + 1 ) ^{4}}{4 t ^{4}} + C

作图

流行的例子

sum from n=1 to infinity of (0.1)^n

n = 1 \sum \infty (0.1)^{n}

derivative of (2x+5/(x^2-3))

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

derivative f(x)=-3x+4

d er i v a t i v e f (x) = - 3 x + 4

斜率 (0,0),(3,4)

s l o p e (0, 0), (3, 4)

derivative-3/(4s^4)+(10)/(3s^3)

d er i v a t i v e - \frac{3}{4 s ^{4}} + \frac{10}{3 s ^{3}}